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IIT - MATHS SET - 4
INDEX 1.
PERMUTATION COMBINATION ......................................................................
2
2.
BINOMIAL THEOREM ......................................................................................
20
3.
CIRCLE ..............................................................................................................
4.
PARABOLA ........................................................................................................
72
5.
ELLIPSE .............................................................................................................
112
6.
HYPERBOLA .....................................................................................................
142
7.
PROBABILITY ...................................................................................................
174
34
IIT- MATHEMATICS-SET-IV
1
Permutation Combination
2
PERMUTATION COMBINATION
FUNDAMENTAL PRINCIPLES OF COUNTING (i)
Multiplication Principle: If an operation can be performed in ‘m’ different ways following which a second operation can be performed in ‘n’ different ways, then the two operations in succession can be performed in m × n ways. This can be extended to any finite number of operations.
Permutation: Definition : Each of the different arrangement which can be made by taking some (or all) number of given things is called a permutation of things. Note : Factorial Notation : The product of first n natural numbers is generally written as n! and is read as factorial n. i.e., n! = n (n – 1)(n – 2) . . . 3. 2. 1 some important properties (i)n! = n (n – 1) !
(ii) (2n)! = 2n. n! [1. 3. 5. 7. . . . (2n – 1)]
(iii)0! = 1! = 1
(iv) factorials of negative integers are not defined
Important Results (i)
The number of permutations of n different things, taking r at a time is denoted by nPr or P (n, r) n! Pr = n r !
n
0 r n
= n (n – 1) (n – 2) . . . (n – r + 1). Note: (i)
The number of permutations of n different things taken all at a time = nPn = n!
(ii)
n
(iii)
n
P0 = 1, nP1 = n and nPn – 1 = nPn = n!. Pr = n (n – 1Pr – 1) = n (n – 1) (n – 2)(n – 2Pr – 2) = n (n – 1) (n – 2) (n – 3 Pr – 3) = . . .
Alternative : (Combinational proof) nPr dentoes the number of ways of arranging r-objects out of n-objects, in a line. This work can be done in the following way also. Suppose the objects are a1, a2, . . ., an. First we find the number of permutations, in which a1 does not appear. Number of such permutations is n – 1Pr – 1 .Further we consider those arrangements, in which a1 necessarily appears. Number of such permutation is r. n – 1Pr – 1, as we can arrange (r – 1) objects out of (n – 1) objects in n – 1Pr– 1 ways, and then in any such permutation we can fix the position of a1 in r - ways. Now using the principle of addition, the required number is n – 1Pr + r. (ii) 3
n–1
Pr – 1 .
The number of permutations of n things taken all at a time, p are alike of one kind, q are alike
IIT- MATHEMATICS-SET-IV of second kind and r are alike of a third kind and the rest n – (p + q + r) are all different is
n! p! q! r !
(iii)
The number of permutations of n different things taken r at a time when each things may be repeated any number of times is nr.
COMBINATION Defination : Each of the different group or selection which can be made by taking some (or all) number of given things, without reference to the order of the things in each group, is called a combination. Important Results (i)
The number of combinations of n different things taken r at a time is denoted by
n Cr or C (n, r) or r
n
n
Cr =
n! r! n r !
0 r n
n
=
=
Pr r!
n n 1 n 2 . . . n r 1 r r 1 r 2 . . . 2.1
If r > n, then nCr = 0. Note: (i)
n
(ii)
n
(iii)
n
Cr is a natural number. C0 = nCn = 1, nC1 = n Cr = nCn – r
(iv) nCr + nCr – 1 = n + 1Cr (v)
n
Cx = nCy x = y or x + y = n
(vi) n. n – 1Cr – 1 = (n – r + 1). nCr –1 n
(vii) If n is even then the greatest value of nCr is n
(viii) If n is odd then the greatest vlaue of nCr is
Cn 2
C n 1 or n C n 1 2
2
.
4
PERMUTATION COMBINATION (ix) nCr =
n n–1 Cr – 1 r
CIRCULAR PERMUTATIONS: (i)
Arrangements round a circular table: Consider five persons A, B, C, D and E to be seated on the circumference of a circular table in order (which has no head). Now, shifting A, B, C, D and E one position in anticlockwise direction we will get arragements as follows: E A
B
A
B
C
D D E
C D
B C
A B
E
C
B
A
E
D
(i)
A (ii)
E
D
(iii)
C (v)
(iv)
we see that arrangements in all figures are same.
The number of circular permutations of n different things taken all at a time is n
Pn (n – 1) !, if clockwise and anticlockwise orders are taken as different. n
DIVISION INTO GROUPS The number of ways in which n distinct objects can be split into r groups containing respectively s1, s2, . . . , sr objects, where si sj , for i j, is given by
n! . s1 !s2 !. . .s r !
If k of the numbers among s1, s2, . . . , sr are equal, then the required number of ways is
n! . s1 !s 2 !. . .s r ! k! For example if 36 distinct objects are to be divided among 9 groups such that four groups have 2 objects each, three groups have 5 objects each and remaining two groups having 6 and 7 objects, then the required number of ways =
36! 4
3
2! .4! 5! .3!.6!.7!
.
DISTRIBUTION AMONG PERSONS The number of ways in which n distinct objects can be distributed among r persons in a required way = number of ways of dividing n distinct objects in r-groups in the required way r!. For example if 36 distinct objects are to be divided among 9 persons such that four of the persons are getting 2 objects each, three of the persons are getting 5 objects each and the remaining two persons are getting 6 and 7 objects, then the number of ways of doing so is
5
IIT- MATHEMATICS-SET-IV
36!
2!
4
3
.4! 5! .3!.6!.7!
9!
.
PRINCIPLE OF INCLUSION AND EXCLUSION If A1, A2, . . . , Am are m sets and n(S) denotes the number of elements in the set S, then
m m s s1 n A k n A k n Ai A j . . . 1 n Ai k 1 i j m i i1 i 2 ... is m k 1 k 1 k 1
1
m 1
. . .
m n Ak k 1 m
Note that if x A k , then x belongs to at least one of Ak, 1 k m. k 1
As another application of the principle of inclusion and exclusion, number of derangement of n objects (number of ways in which n numbered balls (from 1 to n) can be placed in n numbered boxes (from 1 to n), one in each box, so that no ball goes to its corresponding numbered box), denoted by n can be obtained. Infact we have 1 1 1 n 1 n = n 1 1 2 3 . . . 1 n .
MULTINOMIAL THEOREM Consider the equation x1 + x2 + . . . +xr = n, where a i x i bi ; a i, bi , x i I;i 1, 2, . . ., r. In order to find the number of solutions of the given equation satisfying the given conditions we observe that the number of solutions is the same as the coefficient of xn in the product
x
a1
x a1 1 x a1 2 . . .x b1
x
x
a3
x a 3 1 x a3 2 . . .x b3 . . . . . . . x a r x a r 1 x a r 2 . . .x br .
a2
x a 2 1 x a 2 2 . . .x b2
For example, if we have to find the number of non-negative integral solutions of x1 + x2 + . . . + xr = n, then as above the required number is the coefficient of xn in
x0 x1 . . . xn x0 x1 . . . xn . . . x0 x1 . . . xn r brackets = Coefficient of xn in (1 + x + x2 + . . . + xn)r = Coefficient of xn in (1 + x + x2 + . . . )r = Coefficient of xn in (1 – x)– r =Coefficient of xn in
6
PERMUTATION COMBINATION
r r 1 x 2 r r 1 r 2 x 3 . . . 1 r x 2! 3! r r 1 2 r 2 3 = Coefficient of xn in 1 C1x C 2 x C3x . . . n r 1
C n n r 1 Cr 1 .
Note : If there are objects of one kind, m objects of second kind, n objects of third kind and so on; then the number of ways of choosing r objects out of these objects is the coefficient of xr in the expansion of (1 + x + x2 + x3 + . . . + x ) (1 + x + x2 + . . . + xm) (1 + x + x2 + . . . + xn) Further if one object of each kind is to be included, then the number of ways of choosing r objects out of these objects is the coefficient of xr in the expansion of (x + x2 + x3 + . . . + x ) (x + x2 + x3 + . . . + xm) (x + x2 + x3 + . . . + xn) . . .
EXPONENT OF PRIME IN n! Let p be a given prime and n any positive integer, then maximum power of p present in n! is n n n p + 2 3 . .., where [] denotes the greatest integer function. p p n The proof of the above formula can be obtained using the fact that m gives the number of integral multiples of m in 1, 2, . . . , n; for any positive integers n and m.
The above formula does not work for composite numbers. For example if we have to find the 32 32 maximum power of 6 present in 32!, then the answer is not 6 2 . . . 5 , as 5 is the 6 number of integral multiples of 6 in 1, 2, . . . , 32; and 6 can be obtained on multiplying 2 by 3 also. Hence for the required number, we find the maximum powers of 2 and 3 (say r and s) present in 32!. Using the above formula r = 31 and s = 14. Hence 2 and 3 will be combined (to form 6) 14 times. Thus maximum power of 6 present in 32! is 14.
7
IIT-MATHEMATICS-SET IV
1-A
PERMUTATIONS AND COMBINATIONS
8
PERMUTATIONS AND COMBINATIONS
WORKEDOUT ILLUATRATION ILLUSTRATION : 01 A polygon has 44 diagonals, then the number of its sides are a) 11
b) 7
c) 8
d) none of these
Solution : We have 1 n (n 1) n 2 or n 2 3n 88 0 or (n 1!)(n 8) 0 n 44 = C 2 n
n = 11, since n ¹ –8 ILLUSTRATION : 02 If 7 points out of 12 are in the same straight line, then the number of triangles formed is a) 19
b) 185
c) 201
d) none of these
Solution : Numbers of s =
12
C3 7 C 3
12.11.10 7.6.5 = 220 – 35 = 185 1.2.3 1.2.3
ILLUSTRATION : 03 All the letters of the word EAMCET are arranged in all possible ways. The number of such arrangements in which no two vowels are adjacent to each other is a) 360
b) 144
c) 72
d) 54
Solution : Gap Method. Consonants M, C, T in 3! = 6 ways and 4 gaps and 3 vowels (2 alike) in 4
P3 .
1 12 ways = 12 x 6 = 72 2!
ILLUSTRATION : 04 Out of 10 red and 8 white balls, 5 red and 4 white balls can be drawn in number of ways a) 8 C 5 x 10 C 4
b)
10
C5 x 8C 4
c)
18
C9
d) None
Solution : Ans :
10
C5 x 8C 4
ILLUSTRATION : 05 7 men and 7 women are to sit round a table so that there is a man on either side of a women. The number of seating arrangement is a) (7 !)2 9
b) (6!)2
c) 6!. 7!
d) 7!
IIT-MATHEMATICS-SET IV Solution : Ans : 6! 7! ILLUSTRATION : 06 The number of ways in which we can post 5 letters in 10 letter boxes is a) 50
b) 510
c) 105
d) none of these
Solution : We can post the first letter in 10 ways, the second letter in 10 ways and so on. Thus, the number of ways of posting 5 letters in 10 letter boxes is 10 x 10 x 10 x 10 x 10 = 105 ILLUSTRATION : 07 A class has 30 students. The following prizes are to be awarded to the students of this class. First and second in Mathematics; first and second in Physics first in Chemistry and first in Biology. If N denote the number of ways in which this can be done, then a) 400/N c) 8100/N
b) 600/N d) N is divisible by four distinct prime numbers
Solution : First and second prizes in Mathematics (Physics) can be awarded in 30 P2 30 P2 2 ways. First prize in Chemistry (Biology) can be awarded in 30 (30) ways. Therefore, N 30 P2 2 (30 2 ) 30 4 29 2 2 4.3 4.5 2 . 292 Since 400 = 24. 52. 600 = 23. 3. 52 and 8100 = 22. 34. 52 we get N is divisible by each of 400, 600 and 8100. Also N is divisible by four distinct primes, viz., 2,3,5 and 29. ILLUSTRATION : 08 A letter lock consists of three rings marked with 15 different letters. If N denotes the number of ways in which it is possible to make unsuccessful attempts to open the lock, then a) 482/N c) N is product of 4 distinct prime numbers
b) N is product of 3 distinct prime numbers d) none of these
Solution : Since each ring has 15 positions, the total number of attempts that can be made to open the lock is 153. Out of these, there is just one attempt in which the lock will open. Therefore, N = 153–1 = (15 – 1) (152 + 15 + 1) = 2.7.241 clearly, 482|N and N is product of three distinct prime numbers. ILLUSTRATION : 09 The tens digit of 1! + 2! + 3! + …+49! Is a) 1
b) 2
c) 3
d) 4
10
PERMUTATIONS AND COMBINATIONS Solution : We have 1! + 2! + 3! + 4! = 33. Also 5! = 120, 6! = 720, 7! = 5040, 8! = 40320 and 9! = 326880. Thus, the tens digit of 1! + 2! + … + 9! Is 1. Also, note that n! is divisible by 100 for all n ³10, so that tens digits of 10! + 11! + … + 49! is zero. Therefore, tens digit of 1! + 2! +…+49! Is 1 ILLUSTRATION : 10 Four dice are rolled. The number of possible outcomes in which at least one die shows 2 is a) 1296
b) 625
c) 671
d) none of these
Solution : The total number of possible outcomes is 64. The number of possible outcomes in which 2 does not appear on an die is 54, so that tens digit of 10! + 11! + …+49! is zero. Therefore, the number of possible outcomes in which at least one die shows a 2 is 64 – 54 = 1296 – 625 = 671.
11
IIT-MATHEMATICS-SET IV
SECTION A -SINGLE ANSWER TYPE QUESTIONS 1.
How many numbers greater than 1000 but not greater than 4000 can be formed from with the digits 0,1,2,3, 4; repetition of digits being allowed : a) 374
2.
d) 376
b) 225
c) 250
d) 275
The number of ways in which 5 male and 2 female members of a committee can be seated around a round table so that two females are not seated together is : a) 480
4.
c) 370
A candidate is required to answer 6 out of 10 questions which are divided into 2 groups each containing 5 questions and he is not permitted to attempt more than 4 from each group. In how many ways can he make up his choice? a) 200
3.
b) 375
b) 600
c) 720
d) 840
Let Tn denote the number of triangles, which can be formed by using the vertices of a regular polygon of n sides. If Tn1 Tn 21 then n equals. a) 5
5.
x! y! r!
b) 1956
c) 16
d) 64
b)
x y ! r!
c)
x y
Cr
d)
xy
Cr
b) 552
c) 560
d) 1120
b) 136
c) 455
d) 105
The sides AB, BC, CA of a triangle ABC have 3,4 and 5 interior points respectively on them. The total number of triangles that can be constructed by using these points as vertices is a) 220
11.
d) none of the above
The number of integral solutions of x y z 0 with x 5, y 5, z 5 is a) 135
10.
c) 2739
There are 16 points in a plane, no three of which are in a straight line except 8 which are all in a straight line. The number of triangles that can be formed by joining them equals: a) 504
9.
b) 2211
If x, y and r are positive integers, then x C r x C r 1 y C1 x C r 2 y C 2 ..... y C r = a)
8.
d) 4
The number of different signals which can be given from 6 flags of different colours taking one or more at a time is a) 1958
7.
c) 6
The numbers are picked at random from the numbers 1,2,3,…….,100. The number of ways of selecting the two numbers whose product is a multiple of 3 is a) 528
6.
b) 7
b) 204
c) 195
d) 205
There were two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played between themselves proved to exceed by 66 the number of games that the men played with the women. The number of participants is
12
PERMUTATIONS AND COMBINATIONS a) 6 b) 11 12.
22
C112
d)
20
C8
b) 18
c) 9
d) None of these
b) 12
c) 27
d) 63
b) 320
c) 360
d) None of these
b) 12
c) 10
d) 18
A man has 8 children to take them to a zoo. He takes three of them at a time to the zoo as he can without taking the same 3 children together more than once. How many times will he have to go to the zoo? How many times a particular child will go? a) 56,21
19.
c) 16 C 9
If a denotes the number of permutations of x +2 things taken all at a time, b the number of permutations of x things taken 11 at a time and c, the number of permutations of x–11 things taken all at a time such that a = 182bc then the value of x is a) 15
18.
b) 16 C 5
The number of integers greater than a million can be formed by using the digits 2,3,0,3,4,2,3 is a) 240
17.
d) 4 P2 x 6 P3
In an examination, there are three multiple choice questions and each question has 4 choices. Number of ways in which a student can fail to get all answers correct is : a) 11
16.
c) 4 P2 x 4 P3
In a foot ball championship, 153 matches were played. Every team played one match each with each other. The number of teams participating in the championship is : a) 17
15.
b) 4 C 2 x 4 P3
The number of ways in which a team of 11 players can be selected from 22 players including 2 of them and excluding 4 of them is : a)
14.
d) None of these
Two women and three men are to be seated on chairs numbered 1 to 8. First the women choose the chairs from amongst the chairs maked 1 to 4 and then men select the chairs from the remaining. The number of possible arrangements are : a) 4 C 3 x 4 C 2
13.
c) 13
b) 56,36
c) 31,41
d) none of these
Among 2n objects, n are indentical. The number of ways to select n objects out of these 2n objects is
2 n1
C 0 2 n 1 C 1 ....... 2 n 1 C n
a) 2 n
b)
c) the number of proper subsets of a1 , a 2 .......a n
d) None of the above
5
20.
The value of the expression
47
C 4 52 J C 3 is equal to J 1
a)
21.
47
C5
c) 52 C 4
d) 52 C 5
If one quarter of all three element subsets of the set A= a1 , a 2 a3 .......a n contains the element a3 , then n= a) 10
13
b) 52 C 3
b) 12
c) 14
d) 16
22.
IIT-MATHEMATICS-SET IV Six points in a plane be joined in all possible ways by indefinite straight lines and if no two of them be coincident or parallel and no three pass through, the same point (with the exception of the original 6). The number of distinct points of n is equal to a) 105
b) 45 n
23.
If a n r 0
nn 1n 2 8
b) 9 C 4
b)
d) None of these
d) None of the above
b) 4
c) 6
d) 8
b) 135
c) 210
d) 520
b) N is a perfect cube d) N is a perfect 6th Power
The total number of seven digit numbers the sum of whose digits is even is: b) 45 x 10 5
c) 81 x 10 5
d) 9 x 10 5
The number of integral co-ordinates (integral point means both the co-ordinates must be integers) that the exactly in the interior of the triangle with vertices (0,0) (0,2a) and (21,0) is a) 133
32.
c) 12 C 4 4
If N is the number of positive integer of solution of x1 x 2 x3 x 4 770 Then
a) 9 x 10 6 31.
d) 16
nn 1n 2 n 3 nn 1n 2 n 3 c) 6 8
a) N is divisible by 4 distinct primes c) N is a perfect 4th power 30.
c) 48
The number of non-negative integral solutions of the equation x y 3z 33 is a) 120
29.
b) 52
There are three piles of identical red, blue and green balls and each pile contains at least 10 balls. The number of way of selecting 10 balls if twice as many red balls as green balls are to be selected is : a) 3
28.
d) None of the above
There are n straight lines in a plane, no two of which are parallel and no three pass through the same points. Their points of intersection are joined. Then the number of fresh lines thus obtained is a)
27.
1 c) na n 2
Between two junction stations A and B, there are 12 intermediate stations. The number of ways in which a train can be made to stop at 4 of these station so that no two of these halting stations are consecutive is : a) 8 C 4
26.
b) na n
The number of triangles whose vertices are at the vertices of an octagon but none of whose sides happen to come from the sides of the octagon is : a) 24
25.
d) None of these
n 1 r then equals : n n Cr r 0 C r
a) n 1 a n 24.
c) 51
b) 190
c) 233
d) 105
The number of ways in which candidates A1. A2,……….A10 can be ranked if A1 is always above A2 is a) 45 .8!
b) 45. 2!
c) 45
d) 45 + 8!
14
PERMUTATIONS AND COMBINATIONS 33. In a class of 10 students, there are 3 girls A, B,C the number of different ways that they can be arranged in a row such that no two of the three girls are consecutive is : a) 10! 34.
20
C 3 .17 C 4 .13 C5
8! 3!.3!
C 3 . 20 C 4 .13 C 5
b) 290
nn 1 2
15
C 3 17 C 4 13 C 5
d)
17
C 3 20 C 4 13 C 5
d) 920
b) 5
c) 8
d) none of these
b) 230
c) 256
d) 276
b)
5! 8 x C3 3!
c)
5! 6 x C3 3!
d)
8! 6 x C3 5!
b) 64
c) 128
d) 104
b)
n 1n 2 2
c)
n 1n 2
d) None of the above
A student takes tests in three different subjects. Maximum marks of a test in any one subject is 15. The number of ways in which the student can score 15 marks in all the subjects is : a) P(45,15) b) C(45,15) d) None of the above
There are n white and n black balls marked 1,2,3,……n. The number of ways in which we can arranged these balls in a row so that the neighbouring balls are of different colours is a) n!
43.
20
c) 209
c) Coefficient of x 15 in 1 x x 2 ..... x15 3 42.
c)
If n is a natural number, then the number of non negative integral solutions of x y z n is a)
41.
17
The greatest number of points of intersection of 8 straight lines and 4 circles is a) 32
40.
b)
The number of permutations that can be formed by arranging all the letters of the word “NINETEEN’ in which no two E’s occur together is: a)
39.
3!8!3! 5!
A regular polygon has 23 vertices and consequently 23 sides. The number of additional lines need be drawn so that every pair of vertices will be connected is a) 253
38.
d)
The number of positive integers satisfying the inequality n1 C n 2 n 1 C n1 100 is a) 9
37.
7!8! 5!
There are four oranges, five apples and six mangoes in a fruit basket. The number of ways can a person make a selection of fruits among the fruits in the basket ? a) 210
36.
c)
In an election, three wards of a town are convassed by 4, 5 and 3 men respectively. If 20 men volunteer, then the number of ways they can be allotted to different wards is : a)
35.
b) 7 ! 8!
b) 2n!
c) 2(n!)2
d) (2n!)(n!)2
A double decked bus can accommodate 75 passengers, 35 in the upper deck and 40 in the lower deck. The number of ways can the passengers be accommodated if 5 refuse to sit in the upper deck and 8 refuse to sit in the lower deck are
IIT-MATHEMATICS-SET IV a) 44.
62
62
c)
C 27
75
C 3 . 40 C 2
d) None of these
A word is formed by using four letters from the letters of the word “ MATHEMATICS”. In how may ways can it be done if exactly two letters are identical and other two are different : 5 a) 3 x C 2 x
45.
b)
C 25
4! 2!
7 b) 3 x C 2 x
4! 2!
4! 2!
7 c) C 2 x
d) None of the above
If n C r 1 n1 C r 1 n 2 C r 1 ..... 2 n C r 1 2 n1 C r 2 132 n C r then the value of r and the minimum value of n are a) 11,10
46.
b) 4
a b c d ! a!b!c!
21
C7
c) 4 P2 x 6 P3
d) None of the above
b)
a 2b 3c d ! 2 3 a!.b! c!
c)
a 2b 3c d ! a!b!c!
d) None of the above
b)
20
C7
c)
22
C7
d) None of these
b) 2 n m
c) n 12 m
d) n.2 m
The number of ways in which n distinct objects can be put into two different boxes so that no box remains empty is a) 2 n 1
52.
b) 4 C 2 x 4 P3
If x, y, z........are (m+1) distinct prime numbers, the number of factors of x n . y. z....... is : a) mn 1
51.
d) None of these
The number of ways in which an examiner can assign 30 marks out of 8 questions, giving not more than 2 marks to any question is a)
50.
c) 5
A library has a copies of one book, b copies of each of two books, c copies of each of three books and single copies of d books. The total number of ways in which these books can be distributed is a)
49.
d) 12, 13
Eight chairs are numbered 1 to 8. Two women and three men wish to occupy on e chair each. First the women choose the chairs from amongst the chairs marked 1 to 4 and then the men select from the remaining. The number of possible arrangements is a) 6 C 3 x 4 C 2
48.
c) 12,12
1 1 1 If 5 C 6 C 4 C then r equals …… r r r a) 3
47.
b) 11,12
b) n 2 1
c) 2 n 2
d) n 2 2
The adjacent figure is to be coloured using three different colours. The number of ways in which this can be done if no two adjacent triangles have the same colours is a) 81 b) 24 c) 6 d) none of these
16
PERMUTATIONS AND COMBINATIONS
53.
A student is allowed to select at most n books from a collection of 2n +1 books. If the total number of ways in which he can select at least one book is 255, then the value of n is a) 3
54.
b) 4
b) 1512
b) 6 C 6
c)
b) 11
2 m 3
C3
1 m 1 2m 2 4m 3 3
The sum 0 i a) 2 10 1
60.
d) 6
c)12
d)13
b)
1 m 1 2m 2 4m 1 3
d)
1 m 1 2m 2 4m 3 3
b) 601
j 10
10
c) 728
d) none of these
c) 3 10 1
d) 3 10
C j . j C i is equal to b) 2 10
The number of ordered pairs (m,n), m, n {1,2……100} such that 7 m + 7n is divisible by 5 is a) 1250
17
6! 3!3!
In a plane, there are 37 straight lines of which 13 pass through the point A and 11 pass through the point B. Besides no three lines pass through one point, no line pass through both points A and B and no two are parallel. Then the number of intersection points the lines have is equal to a) 535
59.
c)
A person goes in for an examination in which there are four papers with a maximum of m marks from each paper. The number of ways in which one can get 2m marks is a)
58.
d) none of the above
In a certain test, there are n questions. In this test 2 nk students gave wrong answers to at least k questions, where k = 1,2,3,……..n If the total number of wrong answers given is 2047, then n is equal to : a) 10
57.
c) 3024
Note the arrangement (1,1) (1,2) (1,3) (2,3) (3,3),(3,4)(4,4). Here we start from (1,1) then increase one of the co-ordinates by 1 and repeat the same until we reach (4,4). For ILLUSTRATION (1,1)(2,1)(2,2)(2,3)(3,3)(3,4)(4,4) is another such arrangement. The number of such arrangements is a) 6 P6
56.
d)6
The number of ways in which a mixed double game can be arranged from amongst 9 married couples, if no husband and wife play in the same game is a) 756
55.
c) 5
b) 2000
c) 2500
d) 5000
IIT-MATHEMATICS-SET IV 61.
The number of distinct terms in the expansion of x 1 x 2 ........x n 3 is a)
n1
C3
b)
n2
C3
c)
n3
C3
d) none of these
18
PERMUTATIONS AND COMBINATIONS
SECTION B -MULTIPLE ANSWER TYPE QUESTIONS 1.
The number of ways of painting the faces of a cube with six different colours is a) 1
2.
b) 30
d) 8 C 6
c) 6!
The product of r consecutive integers is divisible by r 1
a) r
b)
k
c) r !
d) none of these
k 1
3.
Sanjay has 10 friends among whom two are married to each other. She wishes to invite 5 of the them for a party. If the married couple refuse to attend separately then the number of different ways in which she can invite five friends is a) 8 C 5
4.
10! 2
b) n C n x (m 1)!
c) n C n x m !
d)
n 1
Pn 1
b) 8 ! x 10 C 2
c)
10
P2
d)
10
C2
b) 30
c) 6 C 2
d) 48
b) 25200
c)
10
C4
10! d) 2! 2! 3! 3!
Ten persons, amongst whom are A, B & C are speak at a functional. The number of ways in which it can be done of A wants to speak before B, and B wants to speak before C is a)
19
d) none of these
The number of ways of distributing 10 different books among 4 students (S1 - S4) such that S1 and S2 get 2 books each and S3 and S4 get 3 books each is a) 12600
8.
C 5 2x 8 C 4
In a class tournament when the participants were to play one game with another, two class players fell ill, having played 3 games each. If the total number of games played is 84, the number of participants at the beginning was a) 15
7.
10
The number of ways in which 10 candidates A1, A2 .... A10 can be ranked so that A1 is always above A2 is a)
6.
c)
There are n seats round a table marked 1, 2, 3 .... n. The number of ways in which m (£ n) persons can take seats is a) n Pn
5.
b) 2 x 8 C 3
10 ! 6
b) 21870
10 ! c) 3!
d)
10
P7
IIT-MATHEMATICS-SET-IV
2
BINOMIAL THEOREM
20
BINOMIAL THEOREM
DEFINITION OF BINOMIAL EXPRESSION AND BINOMIAL EXPANSION An expression containing two terms, is called a binomial expression. For example a + b/x, x + 1/y, a – y2 etc. are binomial expressions. Expansion of (x + a)n is called Binomial Expansion. Expression containing three terms are called trinomials. For example x + y + z is a trinomial expression. In general an expression containing more than two terms is called a multinomial. Definition of binomial theorem If n is a positive integer and x, y are two complex numbers, then n
x y n n Cr x n r y r r 0
= nC0xn + nC1xn – 1 y + nC2xn – 2 y2 + . . . + nCn yn
. . . (i)
The coefficients nC0, nC1, . . . , nCn are called binomial coefficients, while (i) is called the binomial expansion. Some important facts regarding Binomial Expansion 1.
There are (n + 1) terms in the expansion.
2.
The sum of the exponents of x and y in any term of the expansion is equal to n.
3.
The binomial coefficients of terms equidistant from the beginning and the end are equal, since nCr = nCn – r .
4.
The term nCr xn – r yr is the (r + 1)th term from the beginning of the expansion. It is usually denoted by Tr + 1 and is called the general term of the expansion.
5.
The rth term from the end is equal to the (n – r + 2)th term from the beginning, i.e., n Cn – r + 1 xr – 1 yn – r + 1 .
6.
n If n is even, then the expansion has only one middle term, the 1 th term i.e., 2 n
Cn / 2 x n / 2 y n / 2 .
n 1 If n is odd, then the expansion has two middle terms, the th term and the 2 n 3 th 2
n i.e., C n 1 / 2 x
term n 1 / 2
y
n 1 / 2
n and C n 1 / 2 x
n 1 / 2
y
n 1 / 2
.
SOME STANDARD EXPANSIONS n
1.
n n n r r Consider the expansion x y Cr x y r 0
21
. . . (i)
IIT-MATHEMATICS-SET-IV If we replace y by – y in equation (i), we have n
n
r
x y n Cr 1 x n r yr
. . . (ii)
r 0
= 2. n
n
n
C0 x n n C1x n 1y n C2 x n 2 y 2 n Cn 1 y n
. . . (ii)
Adding equations (i) and (ii), we have
C0 x n n C 2 x n 2 y2 n C 4 x n 4 y 4
1 n n x y x y 2
. . . (iii)
and substracting equations (ii) from (i) we have, n
C1x n 1y n C 3 x n 3 y 3 n C 5 x n 5 y 5
3.
1 n n x y x y 2
. . . (iv)
Putting x = 1 and y = 1 in equation (i), we have n
C0 n C1 n C 2 n Cn 1 n C n 2 n
. . . (v)
Thus, we see that the sum of the binomial coefficients of (x + y)n is 2n. 4.
Putting x = 1 and y = 1 in equation (iii) and (iv), we have n
5.
C0 n C 2 n C4 2n 1 n C1 n C3 n C5
. . . (vi)
Putting x = 1and replacing y by x in equation (i), we have (1 + x)n = nC0 + nC1x + nC2x2 + . . . + nCnxn
. . . (vii)
Replacing x by – x in equation (vii), we have (1 + x)n = nC0 – nC1 x + nC2x2 – . . . + nCn (– 1)n xn
. . . (viii)
GREATEST BINOMIAL COEFFICIENT The greatest coefficient depends upon the value of n. n
no. of greatest coefficient (s)
Even
1
Odd
2
Greatest coefficient n
n
C n 1 2
C n/2 n
and
C n 1 2
(Values of both these coefficeitns are equal) Clearly greatest binomial coefficient corresponds to the coefficient of middle term.
NUMERICALLY GREATEST TERM OF BINOMIAL EXPANSION (a + x)n = C0 an + C1an – 1 x + . . . Cn – 1 a xn – 1 + Cnxn
22
BINOMIAL THEOREM Tr 1 Tr
n n
Cr
C r 1
x n r 1 x a r a
n r 1 x 1 , for given a, x and n, then r n 1 r a a 1 x
If
n 1 So numerically greatest term will be Tr + 1, where r = 1 a x [ ] denotes the greatest integer function. Note : If
n 1 itsel a 1 x
64 1 64 64 3 . So x , 2 2, Thus, we get | x | 2 and | x | 21 21 21 21
SERIES OF BINOMIAL CO-EFFICIENT Sum of the series by the use of differentiation Gernally we use the method of differentiation when the coefficient of binomial expansion Ck is a polynomial in k Sum of the series by the use of integration Generally we use integration for the series having terms of the form r m
rm
Ck or of the form m 1
Ck . m 1 m 2 . . . m j
Sum of the series by comparing the co-efficients of some power of x in an expansion. In this method we use the fact that coefficient of same power of x in an appropriate identity is the given series. Sum of the series by equating the real and imaginary parts
23
IIT-MATHEMATICS-SET-IV
2-A
BINOMIAL THEOREM
24
BINOMIAL THEOREM
WORKEDOUT ILLUATRATIONS In the expansion of (1+x)43, the coefficients of the (2r+1)th and the (r+2)th terms are equal, then the value of r, is a) 14 1.
2.
b) 15
If (1-x+x2)n = a0+a1x+a2x2+….+a2n x2n then a0+a2+a4+…+a2n equals 1 1 1 a) (3n+1) b) (3n – 1) c) (1-3n) 2 2 2
b) 2n
2n-1
Cn-1
( n 1 )( n 2 ) 2
b) 3rd term
c) 4th term
d) 5th term
b) –2.9C2. 7C3
c) 9C7. 7C4
d) None of these
b)
n-1
Cm-1
c) nCm
d) nCm+1
b) (2n+1). 2n-1Cn
c) 2(n+1). 2n-1Cn
d)
2n-1
b)
n+1
Cr
c)
n+1
Cr+1
Cn+(2n+1).
d) None of these
The remainder of 7103 when divided by 25 is a) 7
25
d)
For 1 r n, the value of nCr + n-1Cr + n –2Cr + …..+ rCr is a) nCr+1
8.
n
The value of C02 + 3C12 + 5C22 + …to (n+1) terms, is (given that Cr nCr a)
7.
c)
The coefficient of xm in : (1+x)m + (1+x)m+1…+(1+x)n, mn is a) n+1Cm+1
6.
1 n +3 2
The coefficient of a3b4c in the expansion of (1+a+b-c)9 is a) 2.9C7. 7C4
5.
d)
The first integral term in the expansion of (is its a) 2nd term
4.
d) 17
The no. of terms in the expansion of (a+b+c)n. nN, is a) 2n
3.
c) 16
b) 25
c) 18
d) 9
2n-1
Cn-1
IIT-MATHEMATICS-SET-IV
SECTION A -SINGLE ANSWER TYPE QUESTIONS 1.
The sum of the coefficients in the expansion of 1 x 3x 2 171 is a) 0
b) 1
c) -1
d) 2 n
2.
3.
1 The term independent of x in the expansion of 1 x x is x n
a) C 02 2C12 3C 22 .... n 1C n2
b) C1 C 2 C 3 ..... C n
c) C 02 C12 C 22 ..... C n2
d) C1 2C 2 3C 3 ..... nC n
The value of
C1 C 3 C 5 ..... is equal to 2 4 6
2n 1 a) n 1 4.
2n 1 c) n 1
2n 1 d) n 1
If coefficients of 2r 1 th and r 2 th terms are equal in the expansion of 1 x 43 then the value of r will be a) 14
5.
2n b) n 1
b) 15
c) 13
d)16
If C1 , C 2 ,.....C n are coefficients in the expansion of 1 x n 1 C1 x C 2 x 2 ....... C n .x n then the value of C12 2C 22 3C 32 ..... 2nC 22n :
6.
b)20
2 n 1
d)5
b) 2n 12 n1 C n
C n1
c) 2n 1.2 n1 C n1
D)
2 n 1
C n 2n 1
2 n 1
C n1
If the second, third and fourth terms in the expansion of a b n are 135, 30 and a) n = 4
9.
c)10
The value of C 02 3.C12 5.C 22 .... to n 1 terms is a)
8.
d) n 2
If the second, third and fourth terms in the expansion of x a n are 240, 720 and 1080 respectively,, then the value of n is : a)15
7.
c) 1n1 .n.2n 1 C n
b) 1n 1 n
a) n 2
b) n = 7
c) n = 6
10 respectively, then 13
d) n = 5
Let k and n be positive integers and put S k 1 k 2 k 3 k .....n k then the value of m 1
C 1 S 1 m1 C 2 S 2 ....... m 1 C m S m is 26
BINOMIAL THEOREM a) 1 n m 1 n 10.
If n, r N and
a) 3 , 3 11.
n 1
b) 1 n 1m 1 n c) 1 n m 1 n m 1
d) 1 n m 1
C r k 3 3 C r1 , then k lies in the interval
n
b) 2 ,
c) [ 3 , )
d)
3 ,2
The number of integral terms in the expansion of 51 / 2 71 / 8 1024 is a) 128
b)129
c)130
d)131
6
12.
1 x log x 1 x1 / 2 If the fourth term in the expansion of is equal to 200 and x >1, then x is equal to
a) 10 13.
b) 10
2
c) 10 4
a, b, c, d are any four consecutive coefficients of any binomial expansion then a) A.P. b) G.P.
14.
If the coefficients of 2nd, 3rd, 4th terms in the expansion of 1 x 2n are in A.P. then : b) 2n 2 9n 7 0
c) 2n 2 9n 7 0
d) 2n 2 9n 7 0
The term independent of x in the expansion of t 1 1x t 1 11 x 1 1 t a) 56 1 t
16.
ab bc cd , , are in a b c
c) H.P. d) Arithmetic geometric progression
a) 2n 2 9n 7 0 15.
d) None of the above
3
1 t b) 56 1 t
3
If the nineth term in the expansion of 3log 3 equals a) log1015
1 t d) 70 1 t 25 x 1 7
3 1 / 8 log 3 5
4
x 1
1
8
is 1 t d) 70 1 t
10
4
is equal to 180 and x >1 then x
b) log515
c) log e 15
d) None of the above
a) 1
b) –1
c) n
d) None of the
above 10
18.
x 1 x 1 The coefficient of the term independent of x in the expansion of 2 / 3 is 1/ 3 1/ 2 x x 1 x x
a) 210 19.
27
b) 105
c) 70
d)112
The total number of dissimilar terms in the expansion of x1 x 2 ..... x n 3 is
IIT-MATHEMATICS-SET-IV a) n 20.
nn 1n 2 c) 6
n 3 3n 2 b) 4
3
n 2 n 1 d) 4
2
The coefficient of x m 0 m n in the expansion of 1 1 x 1 x 2 .... 1 x n is a)
n 1
b) n C m
C m 1
d) n C n m
c) n 1 C m
10
21.
3 3 Coefficient of x in 2 is : 2 x 4
a) 22.
405 226
504 259
c)
450 263
d) None of these
If the coefficient of x 2 and x 3 in the expansion of 3 ax 9 are the same, then the value of a is : a) 9/7
23.
b)
b) 7/9
c) –9/7
d) –7/9
The number of terms in the expansion of x 4 y 3 x 4 y 3 a) 6
b) 7
2
c) 8
is : d) 32
n
24.
1 5 If the 4 term in the expansion of ax is then the values of a and n : x 2 th
a) are
1 ,6 2
b) are 1,3
1 ,3 2
c) are
d) can not be found
8
25.
1 2 If the sixth term in the expansion of 8 / 3 x log 10 x is 5600, then the value of x is x
a) 10 26.
b) 1
q Sq p
c) S p
p Sq q
d) S p S q
b) –24
c) 48
d)-48
If the sum of the coefficients in the expansion of 2 x 2 2x 151 vanishes, then the values of is : a) 2
29.
b) S p
The coefficient of x 7 in the expansion of 1 x 4 1 x 9 is: a) 27
28.
d) none of the above
Let S p and S q be the coefficient of x p and x q respectively in 1 x p q then : a) S p S q
27.
c) 100
b)-1
c) 1
d) -2
If the coefficients of three consecutive terms in the expansion of 1 x n are in the ratio
1 : 7 : 42,
then the value of n is 28
BINOMIAL THEOREM a) 60
30.
b) 0
n r If 1 x C r x r 0
a)
C
d) 1 C1 C 2 C ...... n is equal to 2 3 n 1
2n 1 c) n 1 C
d) 2 n1 1
Cn n 1
1 2 then 1 C 1 C ......1 C
0
1
n 1n1 b) n 1!
n n1 n 1!
c)
n 1n n!
d)
n 1n1 n!
The sum of the all the coefficients of the expansion of x y n is 1024. Then the greatest coefficient in the rth term where r = a) 6
34 .
c) 1/2
2 n 1 1 b) n 1
n 1
n
33.
d)None of these
If 1 x n C 0 C1 x C 2 x 2 ....... C n x n , then C 0
a) 2
32.
c)55
1 10 2 n 10 2 2n 10 3 2 n 10 2 n The value of n n . C1 n . C 2 n . C 3 ...... n is : 81 81 81 81 81 a) 2
31.
b)70
b) 5
c) 4
d) none of these
If the ratio of the 7th term from the beginning to the seventh term from the end in the expansion of x
3 1 1 2 3 is then x is : 6 3
a) 9 35.
b) 6
c) 12
d) None of these
If a be the sum of odd terms and b be the sum of even terms in the expansion of even terms in the expansion of a x n , then 1 x 2 n is equal to a) a 2 b 2
36.
c)
a2 b2 2
d) b 2 a 2
183 7 3 3.18.7.25 The value of 6 is 3 6.243.2 15.181.4 20.27.8 15.9.16 6.3.32 64 a) 10
37.
b) a 2 b 2
b) 1
c) 2
d) 20
If a1 , a 2 , a3 , a 4 are the coefficients of any four consecutive terms in the expansion of 1 x n then
a3 a1 is equal to a1 a 2 a3 a 4 a)
29
a2 a 2 a3
b)
a2 1 . 2 a2 a3
c)
2a 2 a 2 a3
d)
2a3 a 2 a3
IIT-MATHEMATICS-SET-IV 38.
The coefficient of x 99 in x 1 x 3 x 5........ x 199 is a) 1+2+3+…..+99
39.
b)1+3+5+…..+99
If f x x n then the value of f 1
c)1.3.5……99
f 1 x f 2 1 f n 1 ...... where f 1 2! n!
d) None of the above r
x denotes the rth order
derivative of f x with respect of x is a) n 40.
b) 2 n
If 6 6 14
2 n 1
b) 20 2 n1
12
C6 3
b)
12
C6 1
C 1C 2 .........C n
a)
12
C6
d)
12
C6 2
is equal to
nn b) n 1!
n 1n n!
n 1 n c) n 1!
d) none of these
n n2
b)
n1 n x n n1
c)
n n4 x n4 n
d) none of these
y ..... y If x, y and r are positive integers, then xCr x Cr1 .y C 1 xC C C equals to r2 2 r
a) 45.
c)
If n is an even positive integer and x > 0, then the condition that the greatest term in the expansion of (1+x)n may have the greatest co-efficient also, is a)
44.
d) None of these
If C 0 , C 1 , C 2 …… C n are co-efficients in the expansion of (1+x)n then the value of
C 0 C 1 C 1 C 2 ......C n 1 C n
43.
c) 25 n
The coefficient of t 24 in 1 t 2 12 1 t 12 1 t 24 is a)
42.
d) None of these
m and f is the fractional part of m, then fm is equal to
a) 15 n1 41.
c) 2 n1
x!.y! r!
b)
x y ! r!
c) x y Cr
d) xy Cr
If 1 x x 2 C 0 C 1 x C 2 x 2 ........ then the value of n
C 0 C 1 C 1 C 2 C 2 C 3 …… is a) 3 n
b) 1n
c) 2 n
d) none of these
30
BINOMIAL THEOREM
SECTION B -MULTIPLE ANSWER TYPE QUESTIONS 1.
If (1+x+x2)n = a 0 a 1 x a 2 x 2 ..... a 2a x 2 n then a) a 0 a 2 a 4 a 6 ..... 0 , if n is odd. b) a 1 a 3 a 5 a 7 ..... 0 , if n is even c) a 0 a 2 a 4 a 6 ..... 0 , if n = 4p, p Î I d) a 1 a 3 a 5 a 7 ..... 0 , if n = 4p+1, p Î I
2.
The value of C 02 3C12 5C 22 ..... to (n+1) terms, is (given that C r n C r ) a)
2n 1
c) 2(n+1), 3.
b) (2n+1).
C n 1 2n 1
d)
Cn
Cn
C n 2n 1,
2n 1
C 2n 1
The number of distinct terms in the expansion of (x+2y-3z+5w-7u)n is a) n + 1 c)
4.
2n 1
2n 1
n4
b) d)
Cn
n4
C4
n 1n 2 . n 3. n 4 24
If n is a positive integer and 3 3 5 2n 1 when a is an integer and 0 < b < 1 then a) a is an even integer b) (a+b)2 is divisible by 22n+1 c) The integer of just below 3 3 5 2 n 1 divisible by 3 d) a is divisible by 10
5.
If (8+3 7 )n = P + F, where P is an integer and F is a proper fraction then a) P is an odd integer c) F (P+F) = 1
6.
b) P is an even integer d) (1-F) (P+F) = 1
If C 0 , C1 , C 2 .....C n are coefficients in the binomial expansion of (1+x)n, then C 0 C 2 C1C 3 C 2 C 4 .,.... C n 2 C n is equal to
2n! a) n 2!n 2! 7.
2n !
2n! b)
n 2!2
c)
In the expansion of (x+y+z)25 a) every term is of the form 25 C r . r C k .x 25 r .y n k .z k b) the coefficient of x 9 y 9 z 9 is 0 c) the number of terms is 331
31
n 2 !2
d)
2n
C n 2
IIT-MATHEMATICS-SET-IV d) none of these n
8.
1 L et a = 1 . Then for each n Î N. n n
a) an ³ 2 9.
b) an < 3
d) an < 2
If (1+2x+3x2)10 = a0 + a1x + a2x2 + ..... + a20x20, then a) a1 = 20
10.
c) an < 4
b) a2 = 210
d) a20 = 22.37.7
c) a4 = 8085
The coefficient of the middle term in the expansion of (1+x)2n is a)
20
b)
Cn
1.3.5....2n 1 n 2 n!
c) 2.6 ... (4n-2)
d) none of these
KEY
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
c
c
b
a
c
d
c
d
b
d
b
16.
b
31.
b
17. 18. 19.
d
a
c
32. 33. 34.
c
a
a
20. 21. 22. 23. 24. 25. 26.
a
d
a
b
a
a
b
35. 36. 37. 38. 39. 40. 41.
a
b
c
b
b
b
d
12. 13. 14. 15.
b
b
b
c
27. 28. 29. 30.
d
c
c
d
42. 43. 44. 45.
a
a
c
d
32
BINOMIAL THEOREM
KEY
1.
a,b
33
2.
3.
4.
5.
6.
7.
8.
9.
10.
bcd abd ad
bc
ad abc abc abc a,b
IIT-MATHEMATICS-SET-IV
3
CIRCLE
34
CIRCLE A circle is the locus of a point which moves in such a way that its distance from a fixed point is constant. The fixed point is called the centre of the circle and the constant distance, the radius of the circle.
Equation of the Circle in Various Forms : *
The simplest equation of the circle is x2 + y2 = r2 whose centre is (0, 0) and radius r.
*
The equation (x – a)2 + (y – b)2 = r2 represents a circle with centre (a, b) and radius r.
*
The equation x2 + y2 + 2gx + 2fy + c = 0 is the general equation of a circle with centre (– g, – f) and radius g 2 f 2 c .
*
Equation of the circle with points P(x1 , y1) and Q(x2, y2) as extremities of a diameter is (x – x1) (x – x2) + (y – y1) (y – y2) = 0.
Equation of a circle under different conditions : CONDITION
EQUATION
(i) Touches both the axes with radius a
(x-a)2 + (y - a)2 = a2
(ii) Touches x-axis only with centre (a, a)
(x– a)2 + (y – a)2 = a2
(iii) Touches y-axis only with centre (a, b)
(x – a)2 + (y – b)2 = a2
Parametric equation of a circle : The equation x = a cosq, y = a sinq are called parametric equations of the circle x2 + y2 = a2 and q is called a parameter. the point (a cosq, asinq) is also referred to as point q. The parametric coordinates of any point on the circle (x – h)2 + (y – k)2 = a2 are given by (h + a cosq, k + a sinq) with 0 £ q < 2p.
Remarks : (i)
Since there are three independent constants g, f, c in the general equation of a circle. x2 + y2 + 2gx + 2fy + c = 0 a circle can be found to satisfy three independent geometrical conditions and no more. For example, when three points on a circle or three tangents to a circle or two tangents to a circle and a point on it are given, the circle can be determined.
(ii)
To find the condition for the general equation of the second degree ax2+2hxy+by2+2gx + 2fy + c = 0 to represent the circle, viz., x2 + y2 + 2gx + 2fy + c = 0 we see that there is no term in xy and that the coefficient of x2 is the same as that of y2 i.e., the coefficient of xy = 0 and coefficient of x2 = coefficient of y2.
(iii)
If an initial line through the centre is assumed, and its point of intersection with the circle is taken as the starting point, then ‘t’ measures the angle made by the radius vector of any point with the initial line. Thus two diametrically opposite points can be taken as (a + a cos t, b + a sin t) and (a + cos (t + p), b + a sin (t + p))
Intercepts made on axes: 35
IIT-MATHEMATICS-SET-IV Solving the circle x2 + y2 + 2gx + 2fy + c = 0 with y = 0 we get, x2 + 2gx + c = 0. If discriminant 4(g2 – c) is positive, i.e., if g2 > c, the circle will meet the x-axis at two distinct points, say (x1, 0) and (x2, 0) where x1 + x2 = – 2g and x1x2 = c. The intercept made on x-axis by the circle = |x1 – x2| = 2 g 2 c . In the similar manner if f 2 > c, intercept made on y-axis = 2 f 2 c .
TANGENTS AND NORMALS Definition : A tangent to a curve at a point is defined as the limiting positions of a secant obtained by joining the given point to another point in the vicinity on the curve as the second point tends to the first point along the curve or as the limiting position of a secant obtained by joining two points on the curve in the vicinity of the given point as both the points tend to the given point. Two tangents, real or imaginary, can be drawn to a circle from a point in the plane. The tangents are real and distinct if the point is outside the circle, real and coincident if the point is on the circle, and imaginary if the point is inside the circle. The normal to a curve at a point is defined as the straight line passing through the point and perpendicular to the tangent at that point. In case of a circle, every normal passes through the centre of the circle.
Chord of Contact : From a point P(x1, y1) two tangents PA and PB can be drawn to the circle. The chord AB joining the points of contact A and B of the tangents from P is called the chord of contact of P(x1, y1) with respect to the circle.
Equations of Tangents and Normals : If S = 0 be a curve than S1 = 0 indicate the equation which is obtained by substituting x=x1and y = y1 in the equation of the given curve, and T = 0 is the equation which is obtained by substituting x2 = xx1, y2 = yy1, 2xy = xy1 + yx1, 2x = x + x1, 2y = y + y1 in the equation S = 0. If S T
x2 + y2 + 2gx + 2fy + c = 0 then S1 º x12 + y12 + 2gx1 + 2fy1 + c, and xx1 + yy1 + g (x + x1) + f (y + y1) + c
*
Equation of the tangent to x2 + y2 + 2gx + 2fy + c = 0 at A(x1, y1) is xx1 + yy1 + g(x + x1) + f (y + y1) + c = 0.
*
The condition that the straight line y = mx + c is a tangent to the circle x2 + y2 = a2 is c2 = a2 (1 + m2) and the point of contact is (a2m/c, a2/c) i.e. y = mx ± a 1 m 2 is always a tangent to the circle x2 + y2 = a2 whatever be the value of m.
*
The joint equation of a pair of tangents drawn from the point A(x1, y1) to the circle 36
CIRCLE x2 + y2 + 2gx + 2fy + c = 0 is T2 = SS1. *
The equation of the normal to the circle x2 + y2 + 2gx + 2fy + c = 0 at any point (x1, y1) lying x x1 y y1 on the circle is x g = y f . 1
1
In particular, equations of the tangent and the normal to the circle x2 + y2 = a2 at (x1, y1) x y are xx1 + yy1 = a2 ; and x = y respectively.. 1
1
*
The equation of the chord of the circle S º 0, whose mid point (x1, y1) is T = S1.
*
The length of the tangent drawn from a point (x1, y1) outside the circle S = 0, to the circle is S1 .
Director Circle: The locus of the point of intersection of perpendicular tangents is called director circle. If ( x ) 2 + ( y ) 2 = r2 is the equation of a circle then its director circle is ( x ) 2 + ( y ) 2 = 2r2
The position of a point with respect to a circle : The point P(x1, y1) lies outside, on, or inside a circle S º x2 + y2 + 2gx + 2fy + c = 0, according as S1 x12 + y12 + 2gx1 + 2fy1 + c > = or < 0. 2m 1 b
p=±
(1 m 2 )
.......(2) 9 7m
or, by means of (1) p = ±
(1 m 2 )
.
Hence, m is given by (9 – 7m)2 = 13(1 + m2) or 18m2 – 63m + 34 = 0 or
(3m – 2)(6m – 17) = 0
The gradients are 2/3 and 17/6; the tangents are y – 10 = or
2 17 (x – 5) and y – 10 = (x – 5) 3 6
2x – 3y + 20 = 0 and 17x – 6y – 25 = 0.
The corresponding normals are y–1=– 37
3 6 (x + 2) and y – 1 = – (x + 2) 2 17
........(3)
IIT-MATHEMATICS-SET-IV or
3x + 2y + 4 = 0 and 6x + 17y – 5 = 0.
The coordinates of the point of contact of the first tangent in (3) are obtained by solving 2x – 3y + 20 = 0 and 3x + 2y + 4 = 0; the coordinates are (–4, 4). Similarly, the point of contact of the second tangent is (7/5, – 1/5)
RADICALAXIS The radical axis of two circles is the locus of a point from which the tangent segments to the two circles are of equal length.
Equation to the Radical Axis In general S – S’ = 0 represents the equaion of the radical Axis to the two circles i.e. 2x(g –g’ ) + 2y (f – f’) + c – c¢ = 0 where S= x2 + y2 + 2gx + 2fy + c = 0 and S’=º x2 + 2 y + 2 g ’ x +2f’y + c’ = 0 *
If S = 0 and S’ = 0 intersect in real and distinct point then S – S’ = 0 is the equation of the common chord of the two circles.
*
If S = 0 and S’ = 0 touch each other, then S – S’ = 0 is the equation of the common tangent to the two circles at the point of contact.
(-g,-f)
(-g, -f)
(-g’,-f’ )
(-g’,-f ’) Common tangent
Common chord
The radical axes of three circles, taken in pairs, are concurrent Let the equations of the three circles S1, S2 and S3 be S1
x2 + y2 + 2g1x + 2f1y + c1 = 0,
S2
x2 + y2 + 2g2x + 2f2y + c2 = 0
and S3
x2 + y2 + 2g3x + 2f3y + c3 = 0. ........(1)
Now, by the previous section, the radical axis of S1 and S2 is obtained by subtracting the equations of these circles ; hence it is S1 – S2 = 0 .......(2) Similarly, the radical axis of S2 and S3 is S2 – S3 = 0 .......(3) The lines (2) and (3) meet at a point whose coordinates say, (X, Y) satisfy 38
CIRCLE S1 – S2 = 0 and S2 – S3 = 0; hence the coordinates (X, Y) satisfy (S1 – S2) + (S2 – S3) = 0; that is , (X, Y) satisfy S1 – S3 = 0
........(4)
But (4) is the radical axis of the circles S1 and S3 and hence the three radical axes are concurrent. The point of concurrency of the three radical axes is called the radical centre.
FAMILY OF CIRCLES : *
If S º x2 + y2 + 2gx + 2fy + c = 0 and S¢ º x2 + y2 + 2g¢x + 2f¢y + c¢ = 0 are two intersecting circles , then S + lS¢ = 0, l ¹ – 1, is the equation of the family of circles passing through the points of intersection S = 0 and S¢ = 0.
*
If S º x2 + y2 + 2gx + 2fy + c = 0 is a circle which is intersected by the straight line µ º ax + by + c = 0 at two real and distinct points, then S + lµ = 0 is the equation of the family of circles passing through the points of intersection of S = 0 and µ = 0. If µ = 0 touches S = 0 at P, then S = lµ = 0 is the equation of the family of circles, each touching µ = 0 at P.
*
The equation of a family of circles passing through two given points (x1, y1) and (x2, y2) can be written in the form. x y 1
x1 y1 1 = 0 where l is a parameter.. x 2 y2 1 The equation of the family of circles which touch the line y - y1 = m(x – x1) at (x1, y1) for any value of m is (x – x1)2 + (y – y1)2 + l(y - y1 – m (x – x1)) = 0. (x–x1) (x– x2) + (y – y1)(y – y2) + l
*
THE CONDITION THAT TWO CIRCLES SHOULD INTERSECT A necessary and sufficient condition for the two circles to intersect at two distinct points is r1 + r2 > C1C2 > |r1 – r2|, where C1, C2 be the centres and r1, r2 be the radii of the two circles.
External and Internal Contacts of Circles : If two circles with centres C1(x1, y1) and C2(x2, y2) and radii r1 and r2 respectively, touch each other externally, C1C2 = r1 + r2. Coordinates of the point of contact are
r1x 2 r2 x1 r1y 2 r2 y1 A º r r , r r . 1 2 1 2
39
IIT-MATHEMATICS-SET-IV The circles touch each other internally if C1C2 = r1– r2. Coordinates of the point of contact are
r1x 2 r2 x1 r1 y 2 r2 y1 T º r r , r r . 1 2 1 2
Common Tangents to Two Circles : (a) The direct common tangents to two circles meet on the line of centres and divide it externally in the ratio of the radii. (b) The transverse common tangents also meet on the line of centres and divide it internally in the ratio of the radii.
Note : (i)
When two circles are real and non-intersecting, 4 common tangents can be drawn.
(ii)
When two circles touch each other externally, 3 common tangents can be drawn to the circle.
(iii)
When two circles intersect each other, two common tangents can be drawn to the circles.
(iv)
When two circles touch each other internally 1 common tangent can be drawn to the circles.
ORTHOGONAL CIRCLES Two circles are said to be orthogonal if the tangents to the circles at either point of intersection are at right angles. A r2
r1 Q1 (-g,-f)
Q2 (-G,-F)
B
In fig. Q1 and Q2 are the centres of the circles S1
x2 + y2 + 2gx + 2fy + c = 0
S2
x2 + y2 + 2Gx + 2Fy + C = 0 ........(2)
.......(1)
the circles, S1 and S2, intersect at A and B. The tangent at A to the circles S1 is perpendicular to the radius Q1A, and the tangent at A to S2 is perpendicular to the radius Q2A. Hence, if the two tangents are at right angles, then the radii 40
CIRCLE Q1A and Q2A must also be at right angles. Accordingly, the condition that S1 and S2 should be orthogonal is that ÐQ1AQ2 should be 90º; by Pythagoras’ theorem this condition is equivalent to Q1Q22 = Q1A2 + Q2A2 = r12 + r22
.......(3)
(g – G)2 + (f – F)2 = g2 + f2 – c + G2 + F2 – C or, on simplification, 2(gG + fF) = c + C.
.......(4)
Since, AQ2 is perpendicular to the radius Q1A, the tangent at A to the circle S1 passes through the centre of the circle S2; similarly, the tangent at A to S2 passes through the centre of S1. In numerical examples the procedure of solution should be based on the condition expressed by (3).
41
IIT-MATHEMATICS SET IV
3-A
CIRCLES
42
CIRCLES
WORKEDOUT ILLUATRATION ILLUSTRATION : 01 If 2,6 is an interior point of the circle x 2 y 2 8x 12 y p 0 and the circle neither cuts nor touches any one of the axes of co-ordinates then A) p (36,47)
B) p (16, 47)
C) p (16, 36)
D) none of these
Solution : We have x 2 y 2 8x 12 y p 0 then centre and radius of the circle are (4,6) and respectively.
52 p
Circle neither cuts nor touches any one of the axes of coordinates then i.e., 4 52 p
x coordinates of centre > radius p 36 &
……………..(1)
y-coordinate of centre > radius
6 52 p ,
p 16
…………….(2)
D is interior point of the circle then CD < radius
5 52 p
p 47
…………(3)
from (1), (2) & (3) we obtain
36 p 47 p (36, 47).
ILLUSTRATION : 02 The abscissas of two points A and B are the roots of the equations x 2 2ax b 2 0 and their ordinates are the roots of the circle with AB as diameter is A)
43
a
2
b 2 p 2 q 2 B)
a
2
p2
C)
b
2
q2
D) None of these
IIT-MATHEMATICS SET IV Solution : Let co-ordinates of A and B are (a, b) and ( g, d) respectively. 2a , b 2
and
ap , q 2
Equation of circle with AB as diameter.
x x y y 0 x 2 y 2 x y 0 x 2 y 2 2ax 2py b 2 q 2 09
Radius =
a
2
p2 b2 q 2
=
a
2
b2 p 2 q 2
ILLUSTRATION : 03 Let (x, y) = 0 be the equation of a circle. If (0, )=0 has equal roots 2,2 and has roots
(, 0)=0
4 ,5 then the centre of the circle is 5
29 A) 2, 10
29 B) ,2 10
29 C) 2, 10
D)None of these
Solution : Let f x , y x 2 y 2 2gx 2fy c 0 0, 0 2 0 2f c 0 have equal roots.
then 2 2
2f c & 2.2 1 1
2 f 2 & c 4 & ,0 0 2g 0 c 0
2 2g c 0 Here c 4 2 2g 4 0
have roots
4 5 2g 5
g
4 ,5 5
29 10
29 Centre = g , f ,2 10
ILLUSTRATION : 04 The locus of the points of intersection of the tangents to the circle x = rcosq, y = rsinq at points whose
44
CIRCLES is 3
parametric angles differ by
A) x 2 y 2 4 2 3 r 2 B) 3 x 2 y 2 1
C) x 2 y 2 2 3r 2
D) 3 x 2 y 2 42
Solution Circle is x 2 y 2 r 2 cos 2 r 2 sin 2
x2 y 2 r 2 Equation of tangent at is
x cos y sin r
…………….(1)
and at is 3 x cos y sin r 3 3
1 1 3 3 x 2 cos 2 sin y 2 cos 2 sin r x cos y sin x 3 sin y 3 cos 2r Þ r 3x sin y cos 2r or x sin y cos
r 3
………….(2)
squaring and adding (1) & (2) then we get
x2 y 2
4r 2 3
Þ 3 x 2 y 2 4r 2
ILLUSTRATION : 05 If one circle of a co-axial system is x 2 y 2 2gx 2fy c 0 and one limiting point is (a, b) then equation of the radical axis will be
A) g a x f b y c a 2 b 2 0
B) 2 g a b 2 0 x 2f g y c a 2 b 2 0
C) 2gx 2fy c a 2 b 2 0
D) None of these
Solution : Given circle S 1 x 2 y 2 2gx 2fy c 0 and let a circle whose limiting point is (a, b)
x 2 y 2 2g' x 2f' y c' 0 Centre of circle (a, b) and radius = 0
45
……………….(1)
IIT-MATHEMATICS SET IV
g'2 f ' 2 c' 0
c' g'2 f'2 Þ c' a 2 b 2 Equation of the second circle is S 2 x 2 y 2 2ax 2by a 2 b 2 0
………………(2)
From (1) and (2), Equation of radical axis is
S1 S 2 0 2g a x 2f b y c a 2 b 2 0 ILLUSTRATION : 06 If t he circle x 2 y 2 2gx 2fy c 0 cuts each of the circles x 2 y 2 4 0,
and
x 2 y 2 2x 4y 2 0 at the extremities of a diameter, then A) c = -4
B) g f c 1
C) g 2 f 2 c 17
D) gf 6
Solution : Let S x 2 y 2 2gx 2fy c 0
S1 x 2 y 2 4 0 S 2 x 2 y 2 6x 8y 10 0 S 3 x 2 y 2 2x 4y 2 0 Common chords are S S 1 2gx 2fy c 4 0
………………..(1)
S S 2 2g 6x 2f 8y c 10 0 …………..(2) S S 3 2g 2x 2f 4y c 2 0
………..(3)
For cutting at the extremities of diameter, chords (1), (2) & (3) pass through the centers of S 1 , S 2 & S 3 respectively, then
c 4 0, 2g 63 + 2f 84 c 10 0 & 2g 2 1 2f 42 c 2 0 after solving c 4, g 2, f 3 ILLUSTRATION : 07 1 t 2 2at x a The locus of the point of intersection of the lines represents ‘t’ being a 1 t 2 and y 1 t parameter. 46
CIRCLES A) Circle
B) Parabola
C) Ellipse
D) Hyperbola
Solution : x2 y 2
a2
1 t 4t a 1 t 2 2
2
2
2 2
x2 y 2 a2 Radical axis or common chord C1 C 2 0 4x 8y 8 0
or
x 2y 2 0 or 2 = 0
Centres of C1 and C 2 are (1,2) and (-1, -2) respectively.. Slope of C1C 2 =
2 2 2 1 1
Slope of ( L 0 ) is
1 2
slope of C1C 2 slope of L 0 1 Hence L is perpendicular to the line joining centers of and . ILLUSTRATION : 08 The equation of a circle is x 2 y 2 4. The centre of the smallest circle touching this circle and the line
x y 5 2 has the coordinates 7 7 , (a) 2 2 2 2
3 3 (b) , 2 2
7 7 , (c) 2 2 2 2
Solution : Here, OB = radius = 2. The distance of (0,0) from x y 5 2 is 5.
the radius of the smallest circle = and OC 2
3 7 2 2
The slope of OA = 1= tan cos
47
1 1 ,sin 2 2
5 2 3 2 2
(d) None of these
IIT-MATHEMATICS SET IV 7 7 , C 0 OC.cos ,0 OC.sin = 2 2 2 2
0,0
O
2 B
C
A
xy 5 2
ILLUSTRATION : 09 The equation of the locus of the middle point of a chord of the circle x 2 y 2 2x y such that the pair of lines joining the origin to the point of intersection of the chord and the circle are equally inclined to the x-axis is A) x y 2
B) x y 2
C) 2x y 1
D) none of these
Solution : Solving y mx and x 2 y 2 2x 2y 0 , we get 21 m x 2 m 2x 2 2x 2mx 0 Þ x 0, 1 m 2 x 2 y 2 2x 2y 0 A
y mx
x
O
B
y mx
Similarly, solving y mx and the equation of the circle, we get x 0,
21 m 2m 1 m A , 1 m 2 1 m 2
21 m 1 m2
21 m 2m 1 m , and B 1 m2 1 m2
Let the middle point of AB be , . Then
48
CIRCLES
1 21 m 21 m 1 21 m 2m1 m 2 2 and 2 1 m 2 1 m2 1 m 1 m 2
2 2m 2 , ; Eliminating m from these, 2 . 1 m 2 1 m2
ILLUSTRATION : 10 Let L1 be a straight line passing through the origin and L 2 be the straight lien x y 1 . If the intercepts made by the circle x 2 y 2 x 3y 0 on and are equal then which of the following equations can represent ? A) x y 0
B) x y 0
C) x 7 y 0
D) x 7 y 0
Solution : Let the line be y
1 m x 2 . 2
1 2 Its distance from the centre 0,0 = 1 m2 2m
1 2m 2 2 12 So, the length of the intercept = 1 m2
2
……………..(1)
4m 2m
The distance of the lien from the other centre (4, 0) =
1 m2
the length of the intercept in this case
2
1 6m 2 2 5 1 m 2
2
………………..(2)
(1) and (2) are equal. Hence, 1
12m 12 4m 12
4 1 m2
4m 12
4 1 m 2
5
4
7m 2 m 1 0 *** 49
12m 12
4 1 m2
1 2
IIT-MATHEMATICS SET IV
SECTION A -SINGLE ANSWER TYPE QUESTIONS 1.
The two conics ax2 + 2hxy + by2 = c and px2 + 2kxy + qy2 = r intersect in four concyclic points, then ab h ab k ab h ab k A) p q k B) p q h C) p q k D) p q h
2.
The values of for which the circle x2 + y2 + 6x + 5 + (x2 + y2 - 8x + 7) = 0 dwindles into a point are A)
3.
34 2 3
B) 2 2
B) cut
D) do not intersect
C) (0, ±5)
D) none
B) (4, -3), (-2, 5)
C) (4, -5, (-2, 3)
D) none
B) x2 + y2 + 4x = 0
C) x2 + y2 - 4y - 12 = 0 D) x2 + y2 + 4x = 12
B) x2 + y2 + 32x + 4y - 235 = 0 D) x2 + y2 + 32x + 4y + 235 = 0
The circle passing through three distinct points (1, k), (k, 1) and (k, k) passes through the points B) (-1, -1)
C) (-1, 1)
D) (1, -1)
If the lines a1 x b1 y c1 =0 and a2 x b2 y c2 0 cut the coordinate axes in concyclic points, then A) a1 a2 b1b2
11.
D) none
The equation of the image of the circle x2 + y2 + 16x - 24y + 183 = 0 by the line mirror 4x+7y+13 = 0 is
A) (1, 1) 10.
1 23 2
C) one lies inside the other
B) (0, ±3)
A) x2 + y2 + 32x - 4y + 235 = 0 C) x2 + y2 + 32x - 4y - 235 = 0 9.
C)
The equation of the circumcircle of the triangle formed by the lines y+x=6, y-x=6 and y = 0 is A) x2 + y2 - 4y = 0
8.
D) none
3x + 4y - 7 = 0 is common tangent at (1, 1) to two equal circles of radius 5. Their centres are the points A) (4, 5), (-2, -3)
7.
43 2 3
Radius of a circle is 5. It cuts x-axis at two points at a distance 3 from the origin. Its centre is A) (0, ± 4)
6.
C)
Two circles x2 + y2 - 4x - 6y - 8 = 0 and x2 + y2 - 2x - 3 = 0 are such that they A) touch
5.
64 2 3
If 2x2 + pxy + 2y2 + (p - 4) x + 6y - 5 = 0 is the equation of a circle, then its radius is A) 2 3
4.
B)
B) a1b1 a2 b2
C) a1 / a2 b1 / b2
D) none
If a circle passes through the points where the lines 3lx - 2y - 1 = 0 and 4x - 3y + 2 = 0 meet the 50
CIRCLES coordinate axes then = A) -1 12.
B) -1/2
C) 1/2
Four distinct points (2, 3), (1, 0), (0, 1) and (0, 0) lie on a circle for A) all integral values of l B) 0 < < 1
13.
B) |a| < 4
C) |a| < 4
D) |a| > 4
B) x + y = 1
C) 2x + 2y - 5 = 0
D) 2x - 2y - 5 = 0
If one end of a diameter of a circle x2 + y2 - 4x - 6y + 11 = 0 is (3, 4), then the other end is A) (-1, -1)
16.
D) only one value of
The end A of diameter AB of a circle is (1, 1) and B lies on the line x + y - 3 = 0. The locus of the centre of the circle is A) x - y = 1
15.
C) < 0
If a chord of the circle x2 + y2 = 8 makes equal intercepts of length a on the coordinate axes then A) |a| < 8
14.
D) 1
B) (1, 2)
C) (4, 3)
D) none
The co-ordinates of A and B are ( x1 , y1 ) and ( x2 , y2 ) and O is the origin. If circles e described on OA, OB as diameters, then length of common chord is
17.
A) ( x1 y2 x2 y1 ) /AB
B) ( x1 y1 x2 y2 ) /AB
C) ( x1 y2 x2 y1 ) /AB
D) ( x1 y1 x2 y2 ) /AB
If the abscissas and ordinates of two points P and Q are the roots of the equations x 2 2ax b 2 =0 and x 2 2 px q 2 =0 respectively, then equation of the circle with PQ as diameter is
18.
A) x 2 y 2 2ax 2 py b 2 q 2 =0
B) x 2 y 2 2ax 2 py b 2 q 2 =0
C) x 2 y 2 2ax 2 py b 2 q 2 0
D) x 2 y 2 2ax 2 py b 2 q 2 0
Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals
A) PQ.RS 19.
PQ RS B) 2
If a straight line through C (- 8 ,
2PQ.RS C) PQ RS
d)
PQ 2 RS 2 2
8 ) making an angle of 1350 with the x-axis cuts the circle x = 5 cos
, y = 5 sin in points A and B, then the length of AB is A) 6 20.
B) 8
C) 10
D) none
A square is inscribed in the circle x2 + y2 - 2x + 4y + 3 = 0. Its sides are parallel to the co-ordinates axes. Then one vertex of the square is A) (1 +
51
2 , -2)
B) (1 - 2 , - 2)
C) (1, -2 + 2 )
D) none of these
IIT-MATHEMATICS SET IV 21.
A square is formed by following two pairs of straight lines y2 - 14y + 45 = 0 and x2-8x+12=0. A circle inscribed in it. The centre of circle is A) (7, 4)
22.
B) (4, 7)
C) (3, 7)
D) (3/8, 4)
A, B, C are three points on the unit circle x2 + y2 = 1 whose parametric angles , and respectively. Through a point P (-1, 0) on the circle chords PA, PB, PC are drawn whose lengths are in G.P. then cos
, cos , cos are in 2 2 2 A) A.P. 23.
24.
B) G.P.
2 2 A) x y
2x 1 =0 3
2 2 B) x y
2x 1 =0 3
2 2 C) x y
2y 1 =0 3
2 2 D) x y
2y 1 =0 3
(-1, 2) is the vertex of an equilateral triangle whose centroid is (1, 1), then the equation of its circumcircle is
B) (-2, 3), 6
B) /3
C) /4
D) /6
B) x2 + y2 - 9x - 9y - 36 = 0 D) x2 + y2 + 9x - 9y + 36 = 0
ABCD is a square of side a. The centre of the circle which circumscribes the square on taking AB and AD as axes is A) (a, -a)
29.
D) none
The vertices of a triangle ABC are the points (6, 0), (0, 6) and (7, 7). The equation of the circle inscribed in the triangle is A) x2 + y2 - 9x - 9y + 36 = 0 C) x2 + y2 - 9x + 9y + 36 = 0
28.
C) (2, 3), 6
The triangle PQR is inscribed in the circle x2 + y2 = 25. If Q and R have co-ordinates (3, 4) and (-4, 3) respectively, then ÐQPR is equal to A) /2
27.
B) x2 + y2 + 2x - 2y - 3 = 0 D) none
If the equation of incircle of an equilateral triangle is x2 + y2 + 4x - 6y + 4 = 0, then the equation of circumcircle of the triangle is A) (-2, -3), 6
26.
D) None of these
Two vertices of an equilateral triangle are (-1, 0) and (1, 0) and its third vertex lies above the x-axis. The equation of its circumcircle is
A) x2 + y2 + 2x + 2y - 3 = 0 C) x2 + y2 - 2x - 2y - 3 = 0 25.
C) H.P.
B) (-a, a)
C) (a/2, a/2)
D) none
The equation of a circle with centre at origin and passing through the vertices of an equilateral triangle whose median is of length 3a is 52
CIRCLES A) x2 + y2 = a2 30.
C) a2/6
D) none
B) 9/10
C) 1/5
D) 1/10 x2 + y2 - 2x
B) (x-9)2 + (y-8)2 = 52C) (x - 7)2 + (y - 3)2 = 52 D) none
Two circles each of radius 5 units touch each other at (1, 2). If the equation of their common tangents is 4x + 3y = 10, then the centres of the two circles are A) (3, 4), (-1, 0)
34.
B) a2 /4
Equation of the circle whose radius is 5 and which touches externally the circle - 4y - 20 = 0 at the point (5, 5) is A) (x-9)2+ (y - 6)2 = 52
33.
D) none
The lines 3x - 4y + 4 = 0 and 3x - 4y - 5 = 0 are tangents to the same circle. The radius of this circle is A) 9/5
32.
C) x2 + y2 = 9a2
A circle is inscribed in an equilateral triangle of side a. The area of any square inscribed in the circle is A) a2 / 3
31.
B) x2 + y2 = 4a2
B) (5, 7), (-3, -3)
C) (5, 5), (-3, 1)
D) none of these
The centre of the two circles each of radius 13 units and having a common tangent 5x + 12y - 17 = 0 at (1, 1) are A) (-6, -13) and (8, 15) B) (6, 13) and (-4, -11) C) (5, 12) and (-3, -10) D) none of these
35.
A variable circle passes through a fixed point A (a, b) and touches the axis of x. Locus of the other end of the diameter through A is A) circle
36.
53
C) 9
D) none
B) circle
C) parabola
D) none
B) straight line
C) ellipse
D) none
The equation x = a cos + b sin and y = a sin - b cos , 0 2 together represent A) parabola
40.
B) 8
The locus of a point which moves such that sum of the squares of its distances from the three vertices of a triangle is constant is A) circle
39.
D) none of these
Locus of a point which moves such that sum of the squares of its distances from the sides of a square of side unity is 9, is A) straight line
38.
C) ellipse
If the two circles x2 + y2 = 9 and x2 + y2 - 8x - 6y +2 = 0 have exactly two common tangents, then the number of integral values of is A) 2
37.
B) parabola
B) straight line
C) ellipse
D) circle
Locus of the centre of the circle which always passes through the fixed points (a, 0) and 0) is
(-a,
IIT-MATHEMATICS SET IV A) x =1 41.
C) x + y = 2a
B) y = 0
C) x + y = 0
B) a pair of two distinct straight lines D) a point
The line joining (5, 0) to (10 cos q, 10 sinq) is divided internally in the ratio 2 : 3 at P. If q varies, then the locus of P is A) a pair of straight lines B) a circle
44.
D) none of these
B) a circle
C) an ellipse
D) a pair of straight lines
If (2, 5) is an interior point of the circle x2 + y2 - 8x - 12y + k = 0 and the circle neither cuts nor touches any one of the axes of co-ordinates, then A) k (36, 47)
46.
C) a straight line
Let AB be a chord of the circle x2 + y2 = r2 subtending a right angle at the centre. Then the locus of the centroid of the triangle PAB as P moves on the circle is A) a parabola
45.
D) x - y = 0
The equation x2 + y2 + 4x + 6y + 13 = 0 represent is A) a circle C) a pair of coincident straight lines
43.
D) x = 0
Circle are drawn through the point (-5, 0) to cut the x-axis on +ive side and making an intercept of 10 units on x-axis. Locus of the centre of such circles is A) x = 0
42.
B) x + y = 6
B) k (16, 47)
C) k (16, 36)
D) none of these
The point (, 4) lies outside the circles S1 = x2 + y2 + 10x = 0 and S2 = x2 + y2 - 12x + 20 = 0 then belong to A) (-, -8) (-2, )
47.
C) (-, -8) (-2, 6) (6, )
B) (-8, -2)
D) none
The point on the circle x2 + y2 - 2x - 4y - 11 = 0 which is farthest from the origin is 8 4 ,2 A) 1 5 5
48.
49.
4 8 ,2 B) 1 5 5
8 4 ,1 C) 2 5 5
D) none
If the equation x cos + y sin = p represents the equation of common chord APQB of the circles x2 + y2 = a2 and x2 + y2 = b2 (a > b) then AP = A)
a 2 p 2 b2 p 2
B)
a 2 p 2 b2 p 2
C)
a 2 p2 b2 p 2
D)
a 2 p2 b2 p 2
The common chord of the circles x2 + y2 - 4x - 4y = 0 and x2 + y2 - 16 = 0 subtends at the origin an angle equal to A) 300
B) 450
C) 600
D) 900 54
CIRCLES 50. A diameter of x2+y2-2x-6y+6=0 is a chord to circle centre (2, 1), then radius of the circle is A) 1 51.
B) parabola
C) circle
D) none
B) /6
C) /2
D) 0
The condition so that the line (x + g) cos + (y + f) sinq = k is a tangent to x2+y2+2gx+2fy+c=0 is A) g2 + f2 = c + k2
54.
D) 4
The angle between two tangents from the origin to the circle (x - 7)2 + (y + 1)2 = 25 is A) /3
53.
C) 3
If the line lx + my = 1 be a tangent to the circle x2 + y2 = a2, then the point (l, m) lies on A) ellipse
52.
B) 2
B) g2 + f2 = c2 + k
C) g2 + f2 = c2 + k2
D) g2 + f2 = c + k
The angle between a pair of tangents drawn from a point T to the circle x2 + y2 + 4x - 6y + 9 sin2a + 13 cos2a = 0 is 2a. The equation of the locus of the point T is A) x2 + y2 +4x - 6y + 4 = 0 C) x2 + y2 + 4x - 6y - 4 = 0
55.
From any point on the circle x2 + y2 = a2 tangents are drawn to the circle x2 + y2 = a2 sin2. The angle between them is A) /2
56.
B) x + 3y = 0
C) 3x - y = 0
D) x + 2y = 0
B) 5
C) 10
D) none of these
B) ff1= gg1
C) f / f1 g / g1
D) none of these
A line meets the co-ordinate axes in A and B. A circle is circumscribed about the triangle OAB. The distances from the end points of the side AB to the line touching the circle at the origin O are equal to p and q respectively. The diameter of the circle is A) p (p + q)
60.
D) none of these
If the two circles x2 + y2 + 2gx + 2fy = 0 and x2 + y2 + 2x + 2y = 0 touch each other, then A) f2 + g2 = f12 + g12
59.
C) 2
The length of the chord of the circle x2 + y2 = 25 joining the points, tangents at which intersect at an angle of 1200 is A) 5/2
58.
B)
If 3x + y = 0 is a tangent to a circle which has its centre at the point (2, -1), then the equation of the other tangent to the circle from the origin is A) x - 3y = 0
57.
B) x2 + y2 + 4x - 6y - 9 = 0 D) x2 + y2 + 4x - 6y + 9 = 0
B) q (p + q)
C) p + q
D) 1/2 (p + q)
P and Q are two symmetrical points about the tangent at origin to the circle x2+y2-x+y=0. If P be (-5, 6), then Q is A) (6, 5)
55
B) (5, 6)
C) (6, -5)
D) (-6, 5)
IIT-MATHEMATICS SET IV 61.
Tangents drawn from the point (4, 3) to the circle x2 + y2 - 2x - 4y = 0 are inclined at an angle. A) /6
62.
B) /4
D)/2
Tangents are drawn to the circle x2 + y2 - 2x - 4y - 4 = 0 from the point (1, 7), then slopes are A) ± 4/3
63.
C) /3
B) ± 3/4
C) 1, 2
D) 3, 0
If a > 2b > 0 then the positive value of m for which y = mx - b 1 m 2 is a common tangent to x2 + y2 = b2 and (x - a)2 + y2 = b2 is 2b
A) 64.
2
a 4b
2
a 2 4b 2 2b
B) 1
b a 2b
C) 2
D) none
B) x2 + y2 - 6x + 16y - 28 = 0 D) x2 + y2 - 6x - 6y - 28 = 0
C) 10 D) 12
A variable circle always touches the line y = x and passes through the point (0, 0). The common chords of above circle and x2 + y2 + 6x + 8y - 7 = 0 will pass through a fixed point whose co-ordinates are A) (1, 1)
68.
D)
Equation of a circle touching the line |x - 2| + |y - 3| = 4 is (x -2)2 + (y-3)2=R2 where R2 = A) 4 B) 8
67.
2b a 2b
The equation of the circle which has a tangent 2x - y - 1 = 0 at (3, 5) on it and with the centre on x + y = 5, is A) x2 + y2 + 6x - 16y + 28 = 0 C) x2 + y2 + 6x + 6y - 28 = 0
66.
C)
The number of tangents that can be drawn from the point (0,1) to the circle x2+y2-2x-4y=0 is A) 0
65.
B)
B) (2, 2)
C) (1/2, 1/2)
D) none
The equation of a circle which has its centre on the positive side of x-axis and cuts off a chord of length 2 along the line - x = 0 and also touches the line y = x is A) x2 + y2 - 4x + 1 = 0 B) x2 + y2 - 4x +2 = 0 C) x2 + y2 - 8x + 8 = 0 D) x2 + y2 - 8x + 4 = 0
69.
The locus of the point of intersection of tangents to the circle x = a cosq, y = a sinq at the points, whose parametric angles differ by p/2, is A) straight line
70.
C) ellipse
D) none
If the tangent from a point P to the circle x2 + y2 = 1 is perpendicular to the tangent from P to x2 + y2 = 3 then the locus of P is a circle of radius A) 4
71.
B) circle
B) 3
C) 2
D) none
The locus of the point of intersection of the perpendicular tangents of x2 + y2 = 4 is A) x2 + y2 = 8
B) x2 + y2 = 12
C) x2 + y2 = 16
D) x2 + y2 = 4
56
CIRCLES 72.
If 1 , 2 be the inclination of tangents with x-axis drawn from the point P to the circle
x2 + y2 = a2,
then the locus of P, if given that cot 1 cot 2 = c is A) c (x2 - a2) = 2xy 73.
B) 6
B) 144
C) 2
D) 1/2
B) 3x2 + 3y2 + 27x - 2fy + 42 = 0 D) x2 + y2 - 2fx - 9y + 14 = 0
B) (-13, -9)
C) (-6, -7)
D) (13, 7)
The equation of the circle which touches both the axes and the straight line 4x + 3y = 6 in the first quadrant and lies below it, is B) x2 + y2 - 6x - 6y + 9 = 0 D) 4 (x2 + y2 - x - 6y) + 1 = 0
The equation of the circle passing through (2, 1) and touching co-ordinate axes is A) x2 + y2 - 2x - 2y + 1 = 0 C) x2 + y2 - 2x - 2y - 1 = 0
80.
D) none of these
Tangent to the parabola y=x2+6 at (1,7) touches the circle x2+y2+16x+12y+c=0 at the point
A) 4x2 + 4y2 - 4x - 4y + 1 = 0 C) x2 + y2 - 6x - y + 9 = 0 79.
D) 8
Circles are drawn through the point (2, 0) to cut intercept of length 5 units on the x-axis. If their centres lie in the first quadrant, then their equation is
A) (-6, -9) 78.
C) 72
B) 1
A) x2 + y2 - 9x + 2fy + 14 = 0 C) x2 + y2 - 9x - 2fy + 14 = 0 77.
C) 2
Two tangents OA and OB are drawn to the circle x2 + y2 + 4x + 6y + 12 = 0 from origin O. The circumradius of triangle OAB is A) 1/2
76.
D) none
If the circle x2 + y2 + 2gx + 2fy + c = 0 is touched by y = x at P such that OP = 6, then the value of c is A) 36
75.
C) c (y2 - a2) = 2xy
The length of the chord joining the points (4 cos, 4 sin) and [4 cos (+600), 4 sin(+ 600)] of the circle x2 + y2 = 16 is A) 4
74.
B) c (x2 - a2) = y2 - a2
B) x2 + y2 + 2x + 2y + 1 = 0 D) x2 + y2 +2x + 2y - 1 = 0
The equation of common tangent to the circles x2 + y2 + 14x - 4y + 28 = 0 and x2 + y2 - 14x + 4y - 28 = 0 is A) x = 7
81.
B) y = 7
D) 2x - 7y + 14 = 0
A circle of radius 5 units touches both the axes and lies in the first quadrant. If the circle makes one complete roll on x-axis along the positive direction of x-axis, then its equation in the new position is A) x2 +y2 +20px - 10y + 100p2 = 0 C) x2 + y2 - 20px -10y + 100p2 = 0
57
C) 7x - 2y + 14 = 0
B) x2 + y2 + 20px + 10y + 100p2 = 0 D) none of these
IIT-MATHEMATICS SET IV 82.
The equation of the circle passing through the intersection of the circles x2 + y2 - 6x + 2y + 4 = 0, x2 + y2 + 2x - 4y - 6 = 0 and having its centre on the line y = x is A) 3 (x2 + y2) - 5x - 5y + 2 = 0 C) x2 + y2 - 2x - 2y + 1 = 0
83.
The equation of the circle which passes through the origin and the points of intersection of the circle x2 + y2 = 4 and the line x + y = 2 is A) x2 + y2 = 4 (x + y)
84.
B) 7(x2 + y2) - 10x - 10y - 12 = 0 D) x2 + y2 - 6x - 6y + 12 = 0
B) x2 + y2 = 2 (x + y)
C) x2 + y2 = 3 (x + y)
D) x2 + y2 = (x + y)
If the two curves ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 and a' x 2 2h' xy b' y 2 2g ' x 2 f ' y c' 0 intersect in four concylic points, then A)
85.
a b a' b' h h'
B)
a b a' b' h h'
C) h (a - b) = h’ (a’+b’) D) h (a+b) = h’ (a’ - b’)
One of the limit point of the coaxial system of circles containing x2 + y2 - 6x - 6y + 4 = 0, x2 + y2 - 2x - 4y + 3 = 0 is A) (-1, 1)
86.
B) (-1, 2)
c 2 ( a b )2
2c 2 ( a b )2
D)
4c 2 ( a b )2
B) x2 + y2 - x + y = 0
C) x2 + y2 + x - y = 0
D) x2 + y2 + x + y = 0
16 5 , B) 7 10
16 53 , C) 5 10
D) none of these
A variable circle is described to pass through the point (a, 0) and touch the line x+y = 0. Locus of the centre of the above circle is A) parabola
91.
C)
The centre of a circle passing through the point (0, 1) and touching the curve y=x2 at (2, 4) is 16 27 , A) 5 10
90.
B) 4c 2 2( a b )2
The intercept on the line y = x by the circle x2 + y2 - 2x = 0 is AB. Equation of the circle with AB as a diameter is A) x2 + y2 - x - y = 0
89.
B) y = 3, 12x + 5y = 39 D) none of these
The length of the common chord of the circles (x-a)2+(y-b)2 = c2 and (x- b)2 + (y - a)2 = c2 is A)
88.
D) (-2, 2)
The two lines through (2, 3) from which the circle x2 + y2 = 25 intercepts chords of length 8 units have equations A) 2x + 3y = 13, x + 5y = 17 C) x = 2, 9x - 11y = 51
87.
C) (-2, 1)
B) ellipse
C) hyperbola
D) none of these
The equation of tangent drawn from origin to the circle x2 + y2 - 2ax - 2by + b2 = 0 are perpendicular if 58
CIRCLES A) a2 = b2 92.
B) a2 + b2 = 1
C) 2a = b
The equation of the circle passing through (2, 0) and (0, 4) and having the minimum radius is A) x2 + y2 + 2x + 4y = 0 C) x2 + y2 - 2x - 4y = 0
93.
2
C) a parabola
D) a hyperbola
B) 1/ 2
C) 1
D) 2 2
B) b, c, a are in GP
C) c, a, b are in GP
D) None
9 B) 0, 5
9 C) 1, 5
9 D) ,1 5
The point on the y axis, where the line segment joining A (1, 0) and B(3, 0) subtends greatest angle is A) (0, 2)
99.
B) a circle
The values of a for which the point (2, +1) is in the interior of the larger segment of the circle - 2x 2y - 8 = 0 made by the chord x - y + 1 = 0 is 9 A) ,0 5
98.
D) (-4, -6)
If the lines ax + by + c = 0 and cx + by + a = 0 meet the coordinate axes in four concyclic points then A) a, b, c are in GP
97.
C) (4, 6)
There exist two circles passing through (1, 2) and touching both the axes. The length of the common chord is A)
96.
B) (-2, -3)
The locus of the feet of the perpendiculars from (1, 2) to the family of lines (a+3b)x-(2a -b) y - (a - 4b) = 0 where a, b R is a) a straight line
95.
B) x2 + y2 - 2x + 4y = 0 D) x2 + y2 + 2x - 4y = 0
A circle cuts the circles x2 + y2 = 4, x2 + y2 - 6x - 8y + 10 = 0 and x2 + y2 + 2x - 4y - 2 = 0 at the ends of diameter. The co-ordinates of its centre are A) (2, 3)
94.
D) 2b = a
B) (0, )
C) (0, )
D) (0, 3/2)
The equation of the circle having the pair of lines as its normals and having the size just sufficient to contain the circle x (x - 4) + y (y - 3) = 0 is x 2 y 2 6 x 3y k 0 the k equals A) -35
B) -45
C) -15
D) -25
100. The equation of the smallest circle which passes through (2, 1) and touches the x axis is A) x2 + y2 - 2x - 2y - 1 = 0 C) x2 + y2 - x - y - 1 = 0
B) x2 + y2 - 2x - 2y + 1 = 0 D) x2 + y2 - x - y + 1 = 0
101. The radius of the largest circle which passes through (1, 2) and (3, 4) and lies completely in the first quadrant is A) 3
59
B) 2
C)
D) none
IIT-MATHEMATICS SET IV 102. The circle x 2 y 2 4x + 4y + 4 = 0 is inscribed in a triangle, whose two sides are along the axes and the locus of the circumcentre of the triangle is xy + x + y + k A) 1
B) -1
x 2 y 2 0 then k equals
C) ± 1
D) 0
103. The abscissae and ordinates of two points A and B are respectively the roots of the quadratic equations f(x) = 0 and g(x) = 0 and xLt A) 2 f(x) + 3 g(y) = 0
f(x) 2 = then the equation of the circle on AB as diameter is g( x ) 3
B) 3 f(x) + 2 g(y) = 0
C) 2 f(x) - 3 g(y)=0
D) 3 f(x) - 2 g(y) = 0
104. A ray of light incident at the point (3, 1) on the tangent at (0, 1) to the circle gets reflected and the reflected ray touches the circle. The equation of the reflected ray is A) 3x - 4y + 13 = 0
B) 3x + 4y - 13 = 0
C) 4x - 3y - 5 = 0
D) 4x - 3y - 13 = 0
105. Tangents AP and AQ are drawn from A (1, 2) to the circle x2 + y2 + 4x + 2y + 1 = 0 then the equation of the line joining the mid points of AP and AQ is A) x + y = 1
B) x + y + 1 = 0
C) 3x + 3y = 2
D) 3x + 3y + 2 = 0
106. The number of distinct chords of the circle x2 + y 2 - 4x - 3y = 0 passing through (4, 3) which are bisected by the x axis is A) 0
B) 1
C) 2
D) 3
107. A set of circles, each of radius 2 have their centres on the circle x2 + y2 = 36 then the points in the set satisfy A) 16 x2 + y2 64
B) 4 £ x2 + y2 81
C) 0 x2 + y2 64
D) 36 x2 + y2 144
108. The number of rational point(s) (a point (a,b) is rational, if a and b both are rational numbers) on the circumference of a circle having center (,e) is A) at most one
B) at least two
C) exactly two
D) infinite
109. If the equation of the one tangent to the circle with center at (2,–1) from the origin is 3x+y=0 then the equation of the other tangent through origin is A) 3x–y=0
B) x+3y=0
C) x–3y=0
D) x+2y=0
110. If (2,5) is an interior point of the circle and the circle neither cuts not touches any one of the axes of co– ordinates then A) p (36,47)
B) p (16,47)
C) p (16,36)
D) None of these
111. AB is diameter of a circle and C is any point on the circumference of the circle. Then A) The area of ABC is maximum when it is isosceles B) The area of ABC is minimum when it is isosceles 60
CIRCLES C) The perimeter of ABC is maximum when it is isosceles D) None of these 112. The centres of a set of circles, each of radius 3, lie on the circle x 2 y 2 25 . The locus of any point in the set is A) 4 x 2 y 2 64 113.
B) x 2 y 2 25
C) x 2 y 2 25
D) 3 x 2 y 2 9
If the two circles x 2 y 2 2gx 2 fy 0 and x 2 y 2 2g1 x 2 f1 y 0 touch each other, then A) f1 g fg1
C) f 2 g 2 f13 g12
B) ff1 gg1
D) None of these
114. Fir distinct points (2k,3k), (1,0), (0,1) and (0,0) lie on a circle for A) all integral values of k C) k < 0
B) 0
115. Two distinct chords drawn from the point (p,q) on the circle , where ¹0, are bisected by the x–axis. Then A) |p|=|q|
B) p2= 8q2
C) p2<8q2
D) p2>8q2
116. The radical center of three circles described on the three sides of a triangle as diameter is A) The orthocentre C) The incentre of the triangle
B) The circumcentre D) The centroid
117. If the abscissas and ordinates of two points P and Q are the roots of the equations x 2 2ax b 2 0 and x 2 2 px q 2 0 respectively, then equation of the circle with PQ as diameter is A) x 2 y 2 2ax 2 py b 2 q 2 0
B) x 2 y 2 2ax 2 py b 2 q 2 0
C) x 2 y 2 2ax 2 py b 2 q 2 0
D) x 2 y 2 2ax 2 py b 2 q 2 0
118. A variable circle always touches the line y=x and passes through the point (0,0). The common chords of the above circle and x 2 y 2 6 x 8 y 7 0 will pass through a fixed point whose coordinates are 1 3 A) , 2 2
1 1 B) , 2 2
1 1 C) , 2 2
D) None of these
119. A variable chord is drawn through the origin to the circle x 2 y 2 2ax 0. The locus of the center of the circle drawn on this chord as diameter is A) x 2 y 2 ax 0
B) x 2 y 2 ay 0
C) x 2 y 2 ax 0
D) x 2 y 2 ay 0
120. If O is the origin and OP, OQ are distinct tangents to the circle x 2 y 2 2gx 2 fy c 0 , the circumcentre of the triangle OPQ is
61
IIT-MATHEMATICS SET IV A) g , f
B) g, f
C) f , g
D) None of these
121. The are bounded by the circles x 2 y 2 r 2 ,r 1,2 and the rays given by 2x 2 3xy 2 y 2 0 , y>0 is A)
sq.units 4
B)
sq.units 2
C)
3 sq.units 4
D) sq.units
122. The shortest distance from the point (2,–7) to the circle x 2 y 2 14x 10 y 151 0 is A) 1
B) 2
C) 3
D) 4
123. A variable circle always touches the line y=x and passes through the point (0,0). The common chords of above circle and will pass through a fixed point, whose coordinates are A) (1,1)
B) (1/2, 1/2)
C) (–1/2, –1/2)
D) None of these
124. The equation of the circle which touches the axis of coordinates and the line
x y 1 and whose 3 4
centre lies in the first quadrant is x 2 y 2 2 x 2 y 2 0 , where is equal to A) 1
B) 2
C) 3
D) 6
125. A line is drawn through a fixed point P (,) to cut the circle at A and B, the PA. PB is equal to 2
A) r 2
B) 2 2 r 2
2
C) r 2
D) None of these
126. If a circle passes through the points of intersection of the coordinate axes with the lines x–y+1=0 and x–2y+3=0, then the value of is A) 2
B) 1/3
C) 6
D) 3
127. If the circle x 2 y 2 2gx 2 fy c 0 cuts each of the circles x 2 y 2 4 0 , and at the extremities of a diameter, then A) c=–4
B) g+f=c–1
C) g 2 f 2 c 17
D) gf 6
2
128. From the point A(0,3) on the circle x 2 4x y 3 0 , a chord AB is drawn and extended to a point M, such that AM = 2AB. An equation of the locus of M is 2
A) x 2 6 x y 2 0
B) x 2 4 x y 3 =0
C) x 2 y 2 8x 6 y 9 0
D) x 2 y 2 6 x 4 y 4 0
129. A region in the x–y plane is bounded by the curve y=
25 x and the line y=0. If the point (a, a+1) 2
lies in the interior of the region then
62
CIRCLES
130.
A) a(–4,3)
B) a(– , –1) 3,
C) a (–1,3)
D) None of these
C1 is a circle of radius 1 touching the x–axis and the y–axis C2 . is another circle of radius > 1 and touching the axes as well as the circle C1 . Then the radius of C2 is A) 3 2 2
B) 3 2 2
C) 3 2 3
D) None of these
131. The coordinates of the center of the circumcircle of the regular hexagon whose two consecutive vertices have the coordinates (–1,0) and (1,0) and which lies wholly above the x–axis ,are. A) x 2 y 2 2 3 y 1 0
B) x 2 y 2 3 y 1 0
C) x 2 y 2 2 3x 1 0
D) None of these
132. If p and q be the longest distance and the shortest distance respectively of the point (–7,2) from any point (,) on the curve whose equation is then GM of p and q is equal to A) 2 11
B) 5 5
C)13
D) None of these
133. For each kN, let Ck denote the circle whose equation is x 2 y 2 k 2 . On the circle Ck , a particle moves k units in the anti-clockwise direction. After completing its motion on Ck , the particle moves to in the radial direction Ck 1 . The motion of the particle continues in this manner. The particle starts at (1,0). If the particle crosses the positive direction of the x–axis for the first time on the circle Cn then n is A) 7
B)6
C)2
D) None of these
134. The number of points with integral coordinates that are interior to the circle x 2 y 2 16 is Is A) 43
B)49
C)45
D)51
135. There are two circles whose equations are and x 2 y 2 8x 6 y n 0, n Z . If the two circles have exactly two common tangents then the number of possible values of n is. A) 2
B)8
C)9
D) None of these
136. The chords of contact of the pair of tangents to the circle x 2 y 2 1 drawn from any point on the line 2x+y=4 pass through the point A) (1/2, ¼)
B) (1/4,1/2)
C) (1,1/2
D) (1/2,1)
137. A tangent to the circle x 2 y 2 1 through the point (0,5) cuts the circle x 2 y 2 4` at at A and B. The tangents to the circle x 2 y 2 4. at A and B meet at C. The coordinates of C are.
63
IIT-MATHEMATICS SET IV 4 8 6, A) 5 5
4 8 6 , B) 5 5
4 8 6 , C) 5 5
D) None of these
138. A ray of light incident at the point (–2,–1) gets reflected from the tangent at (0,–1) to the circle x 2 y 2 1 . The reflected ray touches the circle. The equation of the line along which the incident ray moved is. A)4x–3y+11=0
B) 4x+3y+11=0
C)3x+4y+11=0
D) None of these
139. The equation of a circle C1 is x 2 y 2 4x 2 y 11 0. A circle C2 of radius 1 rolls on the outside of the circle C1. The locus of the center of C2 has the equation. A) x 2 y 2 4x 2 y 20 0
B) x 2 y 2 43x 2 y 20 0
C) x 2 y 2 3x y 11 0
D) None of these
140. The locus of a point such that the tangents drawn from it to the circle x 2 y 2 6 8 y 0 re perpendicular to each other is A) x 2 y 2 6 x 8 y 25 0
B) x 2 y 2 6 x 8 y 5 0
C) x 2 y 2 6 x 8 y 5 0
D) x 2 y 2 6 x 8 y 25 0
141. If a line segment AM=a moves in the plane XOY remaining parallel to OX so that the left end point A slides along the circle x 2 y 2 a 2 , the locus of M is A) x 2 y 2 4a 2 C) x 2 y 2 2ay
B) x 2 y 2 2ax D) x 2 y 2 2ax 2ay 0
142. If the lines a1 x b1 y c1 0 and a2 x b2 y c2 0 cut the coordinate axes in concyclic point, then
A) a1b1 a2 b2 143.
a1 b1 B) a b 2 2
C) a1 a2 b1 b2
D) a1 a2 b1b2
The equation of the image of the circle x 2 y 2 16 x 24 y 183 0
by the line mirror
4 x 7 y 13 0 is
A) x 2 y 2 32x 4 y 235 0
B) x 2 y 2 32x 4 y 235 0
C) x 2 y 2 32x 4 y 235 0
D) x 2 y 2 32x 4 y 235 0
144. The abscissaes of two points A and B are the roots of the equation x 2 2ax b 2 0 and their ordinates are the roots of the equation x 2 2 px q 2 0 . The radius of the circle with AB as diameter is A)
a
2
b 2 p 2 q 2 B)
a
2
p2
C)
b
2
q2
D) None of these
145. A triangle is formed by the lines whose combined equation is given by x y 4 xy 2x y 2 0 .
64
CIRCLES The equation of its circumcircle is A) x 2 y 2 5x 3x 8 0
B) x 2 y 2 3x 5 y 8 0
C) x 2 y 2 2x 2 y 3 0
D) None of these
146. Let (x,y)=0 be the equation of a circle. If (0,)=0 has equal roots =2, 2 and (,0)=0 has roots =4/5,5 then the cenitre of the circle is A) (2,29/10)
B) (29/10,2)
C) (–2,29/10)
D) None of these
147. The locus of the point of intersection of the tangents to the circle x=r cos, y=r sin at points whose parametric angles differ by /3 is
2 2 2 A) x y 4 2 3 r
2 2 2 C) x y 2 3 r
B) 3 x 2 y 2 1 D) 3 x 2 y 2 42
148. A circle of radius 5 units touches both the axes and lies in the first quadrant. If the circle makes one complete roll on x–axis along the positive direction of x–axis, then its equation in the new position is A) x 2 y 2 20 x 10 y 100 2 =0
B) x 2 y 2 20 x 10 y 100 2 0
C) x 2 y 2 20 x 10 y 100 2 0
D) None of these
149. If two circle (x–1)+(y–3)2=r2 and x 2 y 2 8x 2 y 8 0 intersect in two distinct points, then A) 2 < r < 8
B) r < 2
C) r = 2
D) r >2
150. If the distance from the origin of the centers of three circles x 2 y 2 2 ; x c 2 0 i 1,2,3 are in G. P., then the lengths of the tangents drawn to them from any point on the circle x 2 y 2 c 2 are in A) A. P.
B) G. P.
C) H. P
D) None of these
151. If a circle passes through the point (a,b) and cuts the circle x 2 y 2 2 orthogonally, equation of the locus of its center is A) 2ax 2by a 2 b 2 2
B) ax by a 2 2
C) x 2 y 2 2ax 2by a 2 b 2 2 0 D) x 2 y 2 2ax 2by a 2 b 2 2 0 152. , and are parametric angles of three points P, Q and R respectively, on the circle x 2 y 2 1 and A is point (–1,0). If the lengths of the chords AP, AQ and AR are in G. P., then cos/2, cos/2 and cos/ 2 are in A) A. P.
B) G. P.
C) H. P.
D) None of these
153. The equation of the circle passing through (2,0) and (0,4) and having the minimum radius is A) x 2 y 2 20 C) x 2 y 2 4 x 2 y 2 16 0 65
B) x 2 y 2 2x 4 y 0 D) None of these
IIT-MATHEMATICS SET IV 154. The circle x 2 y 2 4 cuts the line joining the points A(1,0) and B(3,4) in two points P and Q. Let
BQ BP then and are roots of the quadratic equation and QA PA A) x 2 2x 7 0
B) 3x 2 2x 21 0
C) 2x 2 3x 27 0
D) None of these
155. The locus of a point which moves such that the sum of the squares of its distances from the three vertices of a triangle is constant, is a circle whose center is at the …. of the triangle. A) orthocentre
B) centroid
C) Circumcentre
D) nine point center
156. The tangents drawn from the origin to the circle x 2 y 2 2 px 2qy q 2 0 are perpendicular if A) p=q
B) p2=q2
C) q=–p
D) p2+q2=1
157. The equation of a circle C1 is x 2 y 2 4. The locus of the intersection of orthogonal tangents to the circle is the curve C2 and the locus of the intersection of perpendicular tangents to the curve C2 is the curve C3. Then A) C3 is a circle C) C2 and C3 are circles with the same centre
B) The are enclosed by the curve C3 is 8 D) None of these
x y 158. The equation of circle which touches the axes of the coordinates and the line 1 and whose 3 4
centre lies in the first quadrant is x 2 y 2 2cx 2cy c 2 0 , where is A) 1
B) 2
C) 3
D) 6
159. A line meets the coordinate axes in A and B. A circle is circumscribed about the triangle OAB. If m and n are the distances of the tangent to the circle at the origin from the points A and B respectively, the diameter of the circle is A) m m n
B) m n
C) n m n
D)
1 m n 2
160. Length of the tangent drawn from any point of the circle x 2 y 2 2gx 2 fy c 0 to the circle x 2 y 2 2gx 2 fy d 0, d c is A)
cd
B)
d c
C)
g f
D)
f g
161. A square is inscribed in the circle x 2 y 2 10x 6 y 30 0 . One side of the square is parallel to y=x+3, then one vertex of the square is A) (3,3)
B) 7,3)
C) (6, 3 – )
D) (6,3+)
162. The maximum number of points with rational coordinates on a circle whose center is ( 3 ,0 ) is 66
CIRCLES A) one
B) two
C)four
D)infinite
163. The intercept on the line y=x by the circle x 2 y 2 2x 0 is AB. The equation of the circle with AB as a diameter is. A) x 2 y 2 x y 0 B) x 2 y 2 x y
C) x 2 y 2 3x y 0 D) None of these
164. The equation of the incircle of the triangle formed by the axes and the line 4x+3y=6 is A) x 2 y 2 6 x 6 y 9 0
B) 4( x 2 y 2 x y ) 1 0
C) 4( x 2 y 2 x y ) 1 0
D) None of these
165. The equation of the circumcircle of an equilateral triangle is x 2 y 2 2gx 2 fy c 0 and one vertex of the triangle is (1,1). The equation of incircle of the triangle is A) 4( x 2 y 2 ) g 2 f 2 B) 4( x 2 y 2 ) 8gx 8 fy ( 1 g )( 1 3g ) ( 1 f )( 1 3 f ) C) 4( x 2 y 2 ) 8gx 8 fy g 2 f 2 D) None of these 166. If (a,b) is a point on the chord AB of the circle, where the ends of the chord are A=(2,–3) and B=(3,2), then A) a 3,2 .b 2,3 C) a 2,2 ,b 3,3
B) a 2,3 ,b 3,2 D) None of these
167. The range of values of a for which the point (a,4) is outside the circles x 2 y 2 10x 0 and x 2 y 2 12x 20 0 is A) ( ., 8 ) ( 2,6 ) ( 6 , ) C) ( ., 8 ) ( 2, )
B)(–8,–2) D) None of these
168. The angle between the pair of tangents from the point (1,1/2) to the circle x 2 y 2 4x 2 y 4 0 is A) cos-1 4/5
B) sin-1 4/5
C) sin-1 3/5
D) None of these
169. A foot of the normal from the point (4,3) to a circle is (2,1), and a diameter of the circle has the equation 2x–y=2. Then the equation of the circle is. A) x 2 y 2 2x 1 0 B) x 2 y 2 2x 1 0 C) x 2 y 2 2 y 1 0 D) None of these 170. The equation of the smallest circle passing through the intersection of the line x=y=1 and circle x 2 y 2 9 is A) x 2 y 2 x y 8 0
B) x 2 y 2 x y 8 0
C) x 2 y 2 x y 8 0
D) None of these
171. The point P moves in the plane of a regular hexagon such that the sum of the squares of its distances from 67
IIT-MATHEMATICS SET IV the vertices of the hexagon is 6a2. If the radius of the circumcircle of the hexagon is r(
B) an ellipse
C) a circle of radius a 2 r 2
D) an ellipse of major axis a and minor axis r.
KEY 1
2
3
4
5
6
7
8
9
10
11
12 13
14
15
A
B
C
B
A
A
C
D
A
A
C
D
C
C
B
16
17
18 19
20
21
22 23
24
25
26
27 28
29
30
A
A
A
C
D
B
B
C
D
B
C
A
C
B
C
31
32
33 34
35
36
37 38
39
40
41
42 43
44
45
B
B
C
B
B
C
B
A
D
D
A
D
B
B
A
46
47
48 49
50
51
52 53
54
55
56
57 58
59
60
C
B
D
C
C
C
D
C
A
B
C
C
D
A
C
68
CIRCLES
KEY 61
62
63 64
65
66
67 68
69
70
71
72 73
74
75
A
B
C
B
A
A
C
D
A
A
C
D
C
C
B
76
77
78 79
80
81
82 83
84
85
86
87 88
89
90
A
A
A
C
D
B
B
D
B
C
A
B
C
91
92
93 94
95
96
97 98
99 100 101 102 103 104 105
B
B
C
B
C
B
D
106 107
C
69
B
B
C
A
D
A
D
C
B
B
A
IIT-MATHEMATICS SET IV
SECTION B - MORE THAN ONE ANSWER TYPE QUESTIONS 1.
Equations of circles which pass through the points (1, -2) and (3, -4) and touch the x-axis is A) x2 + y2 + 6x + 2y + 9 = 0 B) x2 + y2 + 10x + 20y + 25 = 0 C) x2 + y2 - 6x + 4y + 9 = 0 D) none
2.
The points (2, 3), (0, 2), (4, 5) and (0, c) are concyclic if the value of c is A) 2
3.
B) (-1, -12)
C) (1, 12)
D) (6, 10)
B) å sin a =0
C) å tan a =0
D) å cot a =0
An isosceles triangle ABC with vertex at A (a, 0) is inscribed in the circle x2 + y 2 = a2. If the base angles B and C be each 750, then the co-ordinates of B and C are
a 3 a A) 2 , 2 6.
D) 3
If a, b, g are the parameters of points A, B, C on the circle x2 + y2 = a2 and if the DABC be equilateral, then A) å cos a =0
5.
C) 17
A rectangle ABCD is inscribed in the circle x 2 y 2 3x 12 y 2 0 . If the co-ordinates of A and B are (3, -2) and (-2, 0), then the other two vertices of the rectangle are A) (-6, -10)
4.
B) 1
a 3 a B) 2 , 2
a 3 a C) 2 , 2
a 3 a D) 2 , 2
A square is inscribed in the circle x2 + y2 - 10x - 6y + 30 = 0. One side of the square is parallel to y = x + 3, then one vertex of the square is A) (3, 3)
B) (7, 3)
C) (6, 3- 3 )
D) (6, 3+)
7.
The circles x2 + y2 - 4x - 81 = 0, x2 + y2 + 24x - 81 = 0 intersect each other at points A and B ; a line through point A meets one circle at P and a parallel line through B meets the otehr circle at Q. Then the locus of the mid-point of PQ is A) (x + 5)2 + (y + 0)2 = 25 B) (x - 5)2 + (y - 0)2 = 25 2 2 C) x + y + 10x = 0 D) x2 + y2 - 10x = 0
8.
The equation of tangents drawn from the origin to the circle x2+y2 - 2rx - 2hy + h2 = 0 are A) x = 0 C) (h2 - r2) x - 2rhy = 0
9.
A chord AB of circle x2 + y2 = a2 touches the circle x2 + y2 - 2ax = 0. Locus of the point of intersections of tangents at A and B is A) x2 + y2 = (x - a)2
10.
B) y = 0 D) (h2 - r2) x + 2rhy = 0
B) x2 + y2 = (y - a)2
C) x2 = a (a - 2y)
D) y2 = a (a - 2x)
Equations of circle which touch y-axis at (0, 3) and intercepts a length of 8 units on the is A) x2 + y2 + 10x + 6y + 9 = 0 B) x2 + y2 + 6x - 10y + 9 = 0 C) x2 + y2 + 8x + 4y + 2 = 0 D) x2 + y2 - 10x - 6y + 9 = 0
x-axis
70
CIRCLES 11. Let L1 be a straight line passing through the origin and L2 be the straight line x + y =1. If the intercepts made by the circle x2 + y2 - x + 3y = 0 on L1 and L2 are equal, then which of the following equations can represent L1 ? A) x + y = 0 12.
B) x - y = 0
D) x - 7y = 0
The line y = mx + c intersects the circle x2 + y2 = a2 in two distinct real points if A) - a
13.
C) x + 7y = 0
B) C < a
C) - c
D) a < c
If a circle passes through the points of intersection of the co-ordinate axes with the lines y + 1 = 0 and x - 2y + 3 = 0, then the value of l is A) 2
B) 1/3
C) 6
D) 3
KEY
71
1
2
3
4
5
6
7
8
9
10
11
12 13
bc
ac
ab
ab
bd
ab
ac
ac
ad
ad
bc
ab ab
lx -
IIT-MATHEMATICS-SET-IV
4
PARABOLA
72
PARABOLA
CONIC SECTION The locus of a point, which moves so that its distance from a fixed point is always in a constant ratio to its distance from a fixed straight line, not passing through the fixed point is called a conic section.
The fixed point is called the focus.
The fixed straight line is called the directrix.
The constant ratio is called the eccentricity and is denoted by e.
When the eccentricity is unity i.e., e = 1, the conic is called a parabola ; when e < 1 the conic is called an ellipse and when e > 1, the conic is called a hyperbola.
The straight line passing through the focus and perpendicular to the directrix is called the axis of the parabola
A point of intersection of a conic with its axis is called vertex.
STANDARD EQUATION OFA PARABOLA Let S be the focus, ZM the directrix and P the moving point. Draw SZ perpendicular from S on the directrix. Then SZ is the axis of the parabola. Now the middle point of SZ, say A, will lie on the locus of P, i.e., AS = AZ. Take A as the origin, the x-axis along AS, and the y-axis along the perpendicular to AS at A, as in the figure. Y
M Z
A
S
P NX
Let AS = a, so that ZA also a. Let (x, y) be the coordinates of the moving point P. Then MP = ZN = ZA + AN = a + x. But by definition MP = PS MP2 = PS2 So that, (a + x)2 = (x – a)2 + y2. Hence, the equation of parabola is y2 = 4ax.
LATUS RECTUM The chord of a parabola through the focus and perpendicular to the axis is called the latus rectum.
In the figure LSL’ is the latus rectum. Also LSL’ = 2
Note : 73
4a.a = 4a = double ordinate through the focus S.
IIT-MATHEMATICS-SET-IV Any chord of the parabola y = 4ax perpendicular to the axis of the parabola is called double ordinate.
Two parabolas are said to be equal when their latus recta are equal.
2
Four Common forms of a Parabola : y2 = 4ax
y2 = –4ax
x2 = 4ay
Vertex :
(0 , 0) (0, 0) (0, 0) (0, 0)
Focus :
(a, 0) (– a, 0) (0, a) (0, – a)
Equation of the Directrix
Equation of the axis y = 0 y = 0 x = 0 x = 0
Tangent at the vertexx = 0 x = 0 y = 0 y = 0
x=–a x=a
x2 = –4ay
y = – a y =a
Parametric Coordinates Any point on the parabola y2 = 4ax is (at2 , 2at) and we refer to it as the point ‘t’. Here, t is a parameter, i.e., it varies from point to point.
Focal Chord : Any chord to y2 = 4ax which passes through the focus is called a focal chord of the parabola y2 = 4ax . Let y2 = 4ax be the equation of a parabola and (at2, 2at) a point P on it. Suppose the coordinates of the other extremity Q of the focal chord through P are (at12, 2at1).
Then, PS and SQ, where S is the focus (a, 0) have the same slopes.
2at 0 2at 0 = at 2 a 2 at a 1
Hence
tt12 – t = t1t2 – t1 (tt1 + 1)(t1 – t) = 0
t1 = – 1/t, i.e. the point Q is (a/t2, – 2a/t).
The extremities of a focal chord of the parabola y2 = 4ax may be taken as the points t and – 1/t.
Focal Distance of Any Point : The focal distance of any point P on the parabola y2 = 4ax is the distance between the point P and the focus S, i.e. PS.
74
PARABOLA
Thus the focal distance = PS = PM = ZN = ZA + AN = a + x
Position of a Point Relative to the Parabola : Consider the parabola : y2 = 4ax. If (x1, y1) isa given point and y12 – 4ax1 = 0, then the point lies on the parabola. But when y12 – 4ax1 ¹ 0. We draw the ordinate PM meeting the curve in L. Then P will lie outside the parabola if PM > LM, i.e., PM2 – LM2 > 0
Now, PM2 = y12 and LM2 = 4ax1 by virtue of the parabola. Substituting these values in equation of parabola, the condition for P to lie outside the parabola becomes y12 – 4ax1 > 0. Similarly, the condition for P to lie inside the parabola is y12 – 4ax1 < 0.
THE GENERAL EQUATION OF A PARABOLA We shall now obtain the equation of a parabola when the focus is any point and the directrix is any line. Let (h, k) be the focus S and lx + my + n = 0, the equation of the directrix ZM of a parabola. Let (x, y) be the coordinates of any point P on the parabola. Then the relation, PS = distance of P from ZM, gives (x – h)2 + (y – k)2 = (lx + my + n)2/(l2 + m2) (mx – ly)2 + 2gx + 2fy + d = 0
This is the general equation of a parabola. It is clear that second-degree terms in the equation of a parabola form a perfect square. The converse is also true, i.e. if in an equation of the second degree, the second degree terms form a perfect square then the equation represents a parabola, unless it represents two parabola, unless it represents two parallel straight lines.
Note : The general equation of second degree i.e. ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents
75
IIT-MATHEMATICS-SET-IV
a parabola if D ¹ 0 and h2 = ab, D =
a
h g
h g
b f . f c
Special case : Let the vertex be (a, b) and the axis be parallel to the x-axis. Then the equation of parabola is given by (y – b)2 = 4a (x – a) which is equivalent to x = Ay2 + By + C If three points are given we can find A, B and C. Similarly, when the axis is parallel to the y-axis, the equation of parabola is y = A’x2 + B’x + C
TANGENT DRAWN AT A POINT LYING ON A GIVEN PARABOLA : (i)
If P (x1, y1) be a point on the parabola y2 = 4ax, then the equation of the tangent at P is yy1 = 2a (x + x1) i.e
(ii)
T(x,y) = 0
If P(at2, 2at) be any point on the parabola y2 = 4ax, then
slope of the tangent at P =
2a 1 = 2at t
dy 2a dx y
and hence its equation is 1 y – 2at = (x – at2) t
i.e. yt = x + at2
.........(1)
1 If we substitute m for , in equation (1), we have the following result. The equation t
a y = mx + m
........(2)
Point of Intersection of Tangents at ‘t1’ & ‘t2’ Equations of the tangents at the points (at12, 2at1) and (at22 , 2at2) are (by ARt. 7.6) yt1 = x + at12
.........(4)
and yt2 = x + at22
.........(5)
respectively. Solving equations (4) and (5) gives the coordinates of the intersection point of these two tangents as {at1t2, a(t1 + t2)}
Note: (i)
If the tangents at t1 & t2 are at right angles then t1t2 = –1. 76
PARABOLA (ii)
If the chord joining t1, t2 subtends a right angle at the vertex then t1t2 = – 4 Intersection of a Line and a parabola Let the parabola be y2 = 4ax
......(1)
and the given line be y = mx + c
......(2)
Eliminating y from (1) and (2), then (mx + c)2 = 4ax or
m2x2 + 2x (m c– 2a) + c2 = 0
......(3)
This equation is quadratic in x, gives two values of x. Which shows that every straight line will cut the parabola in two points may be real, coincident or imaginary according as Discriminant o f (3) > , = , < 0 i.e. 4(mc – 2a)2 – 4m2c2 > , = , < 0 or
4a2 – 4amc > , = , < 0
or
a > , = , < mc
......(4)
Note : c2 which gives only one value of x and 4a so every line parallel to x-axis cuts the parabola only in one real point. If m = 0 then equation (3) gives – 4ax + c2 = 0 or x =
Condition of Tangency If the line (2) touches the parabola (1), then equation (3) has equal roots Discriminant of (3) = 0
4(mc – 2a)2 – 4m2c2 = 0 – 4amc + 4a2 = 0 c=
a , m ¹ 0 ......(5) m
so, the line y = mx + c touches the parabola y2 = 4ax if c =
a (which is condition of tanm
gency). Substituting the value of c from (5) in (2) then y = mx +
a , m ¹ 0 ......(6) m
Hence the line y = mx +
a will always be a tangent to the parabola y2 = 4ax. m
The point of Contact The point of contact of the tangents at ‘t’ is (at2 , 2at). In terms of slope ‘m’ of the tangent the
77
IIT-MATHEMATICS-SET-IV a 2a point of contact is 2 , , (m ¹ 0). m m
EQUATION OF NORMAL TO THE PARABOLA If P(at2, 2at) be any point on the parabola y2 = 4ax, then
slope of the tangent at P =
2a 1 = . 2at t
dy 2a dx y
Therefore, slope of the normal at P = – t and its equation is y – 2at = – t(x – at2) i.e. y = – tx + 2at + at3
.....(1)
If we substitute m for – t in equation (1), we have the following result. the equation y = mx – 2am – am3
.......(2)
is a normal real or imaginary can be drawn from any point to a given parabola and the algebraic sum of the ordinates of the feet of these three normals is zero. Let equation of a parabola be y2 = 4ax
......(3)
and that of a normal to it be y = mx – 2am – am3
.......(4)
If this normal passes through the point (x1, y1),we have y1 = mx1 – 2am – am3 i.e. am3 + (2a – x1) m + y1 = 0 This equation gives three values of m, real or imaginary, If m1, m2 and m3 be the roots of equation (5), then we have m1 + m2 + m3 = 0. Hence, the sum of the ordinates of the feet of these normals = – 2a(m1 + m2 + m3) = 0
Note: (i)
2 If normal at the point ‘t1’ meets the parabola again at ‘t2’ then t2 = – t1 – t 1
(ii)
If the normals at t1& t2 meet again on the parabola then t1t2 = 2
(iii)
The point of intersection of the normals to the parabola y2 = 4ax at ‘t1’ and ‘t2’ is [2a + a(t12 + t22 + t1t2), at1t2 (t1 + t2)].
RULE FOR TRANSFORMING AN EQUATION FOR THE VARIOUS FORMS OF 78
PARABOLA
THE PARABOLA In all the previous articles on the parabola, all the related propositions have been proved and derived for the particular parabola y2 = 4ax. However, all the results with slight transformations are valid for any parabola. In this Art. If any equation derived for the parabola y2 = 4ax, (a > 0) is given by E(x, y, a) = 0
......(1)
then the same equation for the parabola y2 = – 4ax will be E(x, y, – a) = 0
.......(2)
For the parabola x2 = + 4ay will be E(y, x, a) = 0
........(3)
and for the parabola x2 = – 4ay will be E(y, x, – a) = 0
........(4)
If the coordinates of the vertex b (a, b), then substitute (x – a) and (y – b) for x and y respectively. Using the above rule for the equation of general tangent to the parabola y2 = – 4ax, equation is y = mx –
a , m
.........(5)
to the parabola (y – b)2 = 4a(x – a) is (y – b) = m(x – a) +
a , m
.......(6)
to the parabola x2 = 4ay is x = my +
a m
.......(7)
Similarly, equation of a general normal to the parabola y2 = – 4ax is y = mx + 2am + am3,
.......(8)
to the parabola x2 = 4ay is x = my – 2am – am3
.......(9)
and so on.
EQUATION OF THE CHORD WHOSE MID-POINT IS GIVEN Let P(x1, y1) be a given point and y2 = 4ax be a given parabola 79
......(1)
IIT-MATHEMATICS-SET-IV Equation of any line passing through P(x1, y1) can be written as
y – y1 = m(x – x1)
......(2)
This line meets the given parabola in point A, B whose abscissa are given by the equation {y1 + m(x – x1)}2 = 4ax i.e. {mx + y1 – mx1}2 = 4ax i.e. m2x2 + {2m(y1 – mx1) – 4a}(y1 – mx1)2 = 0 If x2, x3 be the roots of the equation, then according to the given condition, we have x2 + x3 + 2x1 ie.
[\ P(x1, y1) is the mid-point of AB]
4a 2m( y1 mx 1 ) = 2x1 m2
[from equation (3)]
2a givesm = y 1
Putting the above value of m in equation (2) gives the equation of the required chord as (y – y1) y1 = 2a(x – x1) which on rearranging reduces to yy1 – 2a(x + x1) = y12 – 4ax1
.......(4)
CHORD OF CONTACT Let P(x1, y1) be a given point and y2 = 4ax
.......(1)
be a given parabola.
Let us assume the coordinate of the point of contact A and B (see fig.) as (x2, y2) and (x3, y3). Equations of the tangents are A and B (by Art. 6.7) are yy2 – 2a(x + x2) = 0
.......(2)
and yy3 – 2a(x + x3) = 0
.......(3) 80
PARABOLA Since these tangents pass through P(x1, y1), hence its coordinates must satisfy the equations (2) and (3) i.e. y1y2 – 2a(x1 + x2) = 0 .......(4) and y1y3 – 2a (x1 + x3) = 0 .......(5) Observing equations (4) and (5) it can be seen that the equation of AB is yy1 – 2a(x + x1) = 0
........(6)
Since equations (4) and (5) are true, it follows that A(x2, y2) and B(x3, y3) will satisfy equation (6), which proves therefore, that equation (6) represents a straight line passing through A, B and hence the chord of P. The above equation could also have been obtained using the rule given in Art. 6.16 i.e. T(x, y) = 0
.......(7)
PAIR OF TANGENTS FROM A GIVEN POINT TO A GIVEN PARABOLA Let P(x1, y1) be a given point and y2 = 4ax
.......(1)
be a given parabola. Let M(h,k) be any point on either of the tangents from P on to the given parabola. The equation of the straight line joining P and M is (k y1 ) y – y1 = (h x ) (x – x1)i.e. 1
(k y1 ) (hy1 kx1 ) y = x (h x ) + (h x ) .......(2) 1 1
Since this line is a tangent to the given parabola, therefore it must be of the form a y = mx + .......(3) m
Comparing equations (2) and (3), we have (k y1 ) m = (h x ) 1
and
......(4)
(hy1 kx 1 ) a = (h x ) m 1
.......(5)
Eliminating m by multiplying equations (4) and (5), we have a(h – x1)2 = (k – y1)(hy1 – kx1) Putting (x, y) in place of (h, k) gives the equation of the locus of M (i.e. the required pair of tangents) as a(x – x1)2 = (y – y1)(xy1 – yx1) which on rearranging reduces to
81
IIT-MATHEMATICS-SET-IV 2
2 2
2 1
{yy1 – 2a(x + x1)} = (y – 4ax) (y – 4ax1)
SOME STANDARD PROPERTIES OF THE PARABOLA (1)
The portion of a tangent to a parabola intercepted between the directrix and the curve subtends a right angle at the focus : The equation of the tangent to the parabola y2 = 4ax at P(at2, 2at) is ty = x + at2
......(1)
Let (1) meet the directrix x + a = 0 at Q
a(t 2 1) a , Then co-ordinates of Q is t , also focus S is (a, 0) Slope of SP =
Y Q
2at 0 2t = 2 = m1 (say) 2 at a t 1
P(at2,2at) 90º
V
Z
S(a,0)
X
x+a=0
Directrix
at 2 a 0 t2 1 t and Slope of SQ = = = m2 (say) 2t a a
m1m2 = – 1
i.e., SP is perpendicular to SQ i.e., ÐPSQ = 90º. (2)
The tangent at any point P of a parabola bisects the angle between the focal chord through P and the perpendicular from P to the directrix. Let the tangent at P (at2, 2at) to the parabola y2 = 4ax meet the axis of the parabola. i.e. x-axis or y = 0 at T. The equation of tangent to the parabola y2 = 4ax at P(at2, 2at) is ty = x + at2
T(– at2, 0)
ST = SV + VT = a + at2 = a (1 + t2) P(at2,2at)
M T
Z
V
S(a,0)
X
x+a=0
Directrix
Also SP = PM 82
PARABOLA = a + at2 = a (1 + t2)
SP = ST i.e., STP = SPT
But STP = MPT (alternate angles)
(3)
SPT = MPT.
The foot of the perpendicular from the focus on any tangent to a parabola lies on the tangent at the vertex. Equation of tangent at P (at2, 2at) on the parabola y2 = 4ax is ty = x + at2 or
x – ty + at2 = 0 ......(1)
Now the equation of line through S(a, 0) and perpendicular to (1) is
Equation tx + y = ta
or
t2x + ty – at2 = 0
By adding equation (1) and (2), we get x(1 + t2) = 0 or
(\ 1 + t2 ¹ 0)
x=0
Hence the point of intersection of (1) and (2) lies on x = 0 i.e., on y-axis (which is tangent at the vertex of parabola). (4)
If S be the focus of the parabola and tangent and normal at any point P meet its axis in T and G respectively, then ST = SG = SP. Let P (at2, 2at) be any point on the parabola y2 = 4ax, then equation of tangent and normal at P (at2, 2at) are ty = x + at2 and y = – tx + 2at + at3 respectively. Since tangent and normal meet its axis in T and G. Co-odinates of T and G are (– at2,0) and
(2a + at2, 0) respectively.
SP = PM = a + at2 SG = VG – VS = 2a + at2 – a P(at2,2at)
x+a=0
M 90º T
Z
V Directrix
= a + at2 83
S G (a,0)
IIT-MATHEMATICS-SET-IV and ST = VS + VT = a + at Hence (5)
2
SP = SG = ST
If S be the focus and SH be perpendicular to the tangent at P, then H lies on the tangent at the vertex and SH2 = OS. SP where O is the vertex of the parabola Let P(at2, 2at) be any point on the parabola. y2 = 4ax
......(1)
then tangent at P (at2 , 2at) to the parabola (1) is ty = x + at2 It meets the tangent at the vertex i.e., x = 0
Co-ordinate of H is (0, at)
& SP = PM = a + at2 Y
x+a=0
M H O
2
P(at ,2at)
S(a,0)
X
Directrix
OS = a & SH = or
(a 0) 2 (0 at ) 2 =
a 2 a2t 2
(SH)2 = a. {a (1 + t2)} = OS. SP.
REFLECTION PROPERTY OF A PARABOLA The tangent (PT) and normal (PN) of the parabola y2 = 4ax at P are the internal and external bisectors of ÐSPM and BP is parallel to the axis of the parabola and ÐBPN = ÐSPN.
84
PARABOLA
85
IIT- MATHEMATICS-SET-IV
4-A
PARABOLA
86
PARABOLA
WORKEDOUT ILLUATRATION ILLUSTRATION : 01
If P and Q are the points at 12 ,2at 1 and at 22 ,2at 2 and normals at P and Q meet on the parabola y 2 4ax , then t 1 , t 2 equals A) 2
B) -1
C) -2
D) -4
Solution : Normal at ' t 1' is y t 1x 2at 1 at 13
………..(1)
& normal at ' t 2 ‘ is y t 2x 2at 2 at 32
………….(2)
solving (1) & (2)
we get 2a a t 12 t 22 t 1t 2 , at 1t 2 t 1 t 2 which is lie on y 2 4ax
a 2 t 12 t 22 t 1 t 2 2 8a 2 4a 2 t 12 t 22 t 1t 2
t 12 t 22 t 1 t 2 2 8 4 t 12 t 22 t 1t 2
=
8 4 t 1 t 2 2
t 1 t 2 2 { t12t 22 4] 4t1t 2 2
t 1t 2 2 0
t 1t 2 2
t t
1 2
ILLUSTRATION : 02 The points on the axis of the parabola 3y 2 4y 6x 8 0 from which 3 distinct normals can be drawn is given by 19 2 19 7 4 1 A) a , ; a B) a , ; a C) a , ; a D) none of these 9 3 9 9 3 3 Solution : We have
4 8 y y 2x 0 3 3
2 20 y 2x 3 9
2
2
2 4 8 y 2x 0 3 9 3
2 10 y 2 x 3 9
2
Let y
2 Y and 3
x
Y 2 2X
axis of the parabola Y 0 87
10 X 9
2
IIT- MATHEMATICS-SET-IV i.e.
y 2 / 3
2 Let P x1 , on the axis of the parabola. 3
Equation of Normal in terms of slope. 1 Y mX m m 3 2
i.e. y
2 10 1 m x m m3 3 9 2
2 Three normals meet at P x1 , 3
m=0 &
0 x1
10 1 1 m2 Þ 9 2
1 2 19 m x1 2 9
19 m 2 x1 . but 9
Points on the axis of the parabola is where x1 > 19/9.
19 m 0 x1 9
ILLUSTRATION : 03 The triangle formed by the tangent to the parabola y x 2 at the
point whose abscissa is
x0 x0 1, 2 , the and the straight line has the greatest area if x0
A) 0
B) 1
C) 2
D) 3
Solution : Let P x0 , x02 be any point on the parabola Equation of the tangent at P x0 , x02 is xx0
1 y x02 2
2 xx0 y x02 0
Tangent meet the y-axis at T 0, x02 . Hence the area of the triangle PTQ
1 1 PQ QT = x0 2 x02 2 2
which increases in the interval [1,2] and hence is greatest when x0 2 .
88
PARABOLA y
y x 20
Q
P x0 , x02
x
O
T 0, x 20
ILLUSTRATION : 04
1 1 A focal chord of y 2 4ax meets in P and Q. If is the focus, then SP SQ (A) 1/a
(B) 2/a
(C) 4/a
Solution : a SP PM a at 2 , SQ QN a t 2 1 1 SP a 1 t 2
1 t2 SQ a 1 t 2
1 1 1 SP SQ a
M
xa 0
P at 2 ,2at
S a ,0
N
a 2a , t2 t
ILLUSTRATION : 05 89
(D) none of these
IIT- MATHEMATICS-SET-IV P is a point which moves in the x-y plane such that the point P is nearer to the centre of a square than any of the sides. The four vertices of the square are a, a . The region in which P will move is bounded by parts of parabolas of which one has the equation A) y 2 a 2 2ax
B) x 2 a 2 2ay
C) y 2 2ax a 2
D) none of these
Solution : If P x, y then
x 2 y 2 | a x |, x 2 y 2 | a x | , x 2 y 2 |a-y|,
x 2 y 2 |a y|
squaring then we get the region bounded by the curves
2
2
2
2
x2 y 2 a x , x 2 y 2 a x
x2 y2 a y , x2 y2 a y
ILLUSTRATION : 06 Let there be two parabolas with the same axis, focus of each being exterior to the other and the latus recta being 4a and 4b. The locus of the middle points of the intercepts between the parabolas made on the lines parallel to the common axis is a A) straight line if a=b
B) parabola if a b
C) parabola for all a, b D) none of these
Solution : Which choice of axes as shown in the figure, the equations of the parabolas can be taken as
y 2 4a x k and y 2 4b x k . A line parallel to the common axis is y = h
h2 h2 A k , h B k ,h and Then 4b 4a if P , then
1 h2 h2 h2 1 1 k k , h 2 2 4a 4b 4 a b The locus of P is 2 x
y2 b a 4 ab
90
PARABOLA y 2 4a x k
A P O 0,0
B
x
y 2 4bx k
ILLUSTRATION : 07 Let PQ be a chord of the parabola y 2 4 x . A circle drawn with PQ as a diameter passes through the vertex V of the parabola. If ar=20 unit then the coordinates of P are A) (16, 8)
B) (16, -8)
C) (-16, 8)
D) (-16, -8)
Solution : 2t 0 2 ' m ' of PV t 2 0 t t the equation of QV is y = x. 2
16 8 2 Solving it with y 4 x, Q 2 , t t
Now, ar PVQ
1 PV .VQ 20 (given) 2
PV 2 .VQ 2 402 16 2 8 2 2 40 t t
or
t 2t
or
256 4 256 256 64t 2 40 2 t2
t 4, 1 .
2 2
2
P t 2 ,2t V
0,0
Q 91
t2 4 t2
or
or
t
2
1 256 64 402 2 2 t t
16 t 2 1 0
IIT- MATHEMATICS-SET-IV ILLUSTRATION : 08 The point (a, 2a) is an interior point of the region bounded by the parabola y 2 16 x and the double ordinate through the focus. Then a belongs to the open interval. A) a< 4
B) 0 < a < 4
C) 0 < a < 2
D) a > 4
Solution : (a, 2a) is an interior point of y 2 16 x 0 2
if 2a 16a 0, i.e. a 2 4a 0 V (0,0) and (a, 2a) are on the same side Of x 4 0 , So, a -4 < 0, i.e., a 4 Now, a 2 4a 0
0a4
y 2 16x V
x4
a ,2a
S focus
ILLUSTRATION : 09 The length of a focal chord of the parabola y 2 4ax at a distance b from the vertex is c. Then A) 2a 2 bc
B) a3 b 2 c
C) ac b2
D) b 2 c 4a 3
Solution : Let the ends of the focal chord be at12 , 2at1 and at22 , 2at2 . Then as in t1t2 1 0 .
2
2
Then c 2 a t12 t22 2a t1 t2
2at1 2at2 2 The equation of the focal chord is y 2at1 at 2 at 2 x at1 1 2
or
So,
y 2at1
2 x at12 t1 t2
2 at12 2at1 t t 2at1t2 b 1 2 2 4 4 t1 t2 1 2 t1 t2 92
PARABOLA 2a
b
2 t12 t22
2 2 Also, c a t1 t2
2
t t 4 a t 2
1
2
2
2 1
t22 2 t12 t22 2
c a 2 t12 t22
2 Hence b
a 2 4a 2 4a 3 2 t12 t22 c / a c
ILLUSTRATION : 10 The locus of the middle points of chords of a parabola which subtend a right angle at the vertex of the parabola is A) a circle
B) an ellipse
C) a parabola D) none of these
Solution : Here,
at12 at22 2at1 2at2 , a t1 t2 2 2
2at1 0 2at2 0 Also, at 2 0 at 2 0 1 t1t2 4 1 2 2
2 2 t12 t22 t1 t2 2t1t2 = 2. 4 Now, a a The locus of , is
2 x y2 8 a a2
which is a parabola. ***
at
0,0
at
93
2 1 ,2at 1
y 2 4ax
,
2 2 ,2at 2
IIT- MATHEMATICS-SET-IV
SECTION A -SINGLE ANSWER TYPE QUESTIONS 1.
The co-ordinates of a point on the parabola y2 = 8x whose focal distance is 4 is A) (2, 4)
2.
3
B) 4a/ 2
C) 2a 2
D) None of these
B) (4a, - 4a)
C) (0, 0)
D) (8a, 0)
B) (1, 2)
C) (3/4, 1)
D) (5/4, 1)
B) focus (0, 6)
C) latus rectum 4
D) axis y = 6
B) (-2, 0)
C) (4, 0)
D) (-4, 0)
B) (-1, 6)
C) (1, -6)
D) (-1, -6)
The equation of the directrix of the parabola y2 + 4y + 4x + 2 = 0 is A) x = -1
11.
D) a cosec2
The two ends of latus rectum of a parabola are the points (3, 6) and (-5, 6). The focus is A) (1, 6)
10.
C) a sec2
If (2, 0) is the vertex and y-axis the directrix of a parabola, then its focus is A) (2, 0)
9.
B) 4a sin / cos2
The equation of parabola is given by y2 + 8x - 12y + 20 = 0. Tick the correct options given below A) vertex (2, 6)
8.
D) 8a 3
The focus of the parabola y2 - x - 2y + 2 = 0 is A) (1/4, 0)
7.
C) 6a 3
A square has one vertex at the vertex of the parabola y2 = 4ax and the diagonal through the vertex lies along the axis of the parabola. If the ends of the other diagonal lie on the parabola, the co-ordinates of the vertices of the square are A) (4a, 4a)
6.
B) 4a 3
In the parabola y2 = 4ax, the length of the chord passing through the vertex and inclined to the axis at an angle p/4 is A) 4a 2
5.
D) (4, -2)
In the parabola y2 = 4ax, the length of the chord passing through the vertex and inclined to the x-axis at an angle is A) 4a cos / sin2
4.
C) (2, -4)
An equilateral triangle is inscribed in the parabola y2 = 4ax whose vertex is at the vertex of the parabola. The length of its side isi0073 A) 2a
3.
B) (4, 2)
B) x = 1
C) x = -3/2
D) x = 3/2
If the line x-1=0 is the directrix of the parabola y2-kx + 8 = 0, then one of the values of k is A) 1/8
B) 8
C) 4
D) 1/4
94
PARABOLA 12. If ax2 + 4xy + y2 + ax + 3y + 2 = 0 represents a parabola, then a = A) -4 13.
B) x = -a/2
C) x = 0
B) y2 = 4x - 8
D) x = a/2
C) x2 = 4x - 3
D) none
The equation y2 - 2x - 2y + 5 = 0 represents A) circle centred at (1, 1) C) parabola with focus at (1, 2)
16.
D) none
The equation of the parabola whose vertex is (2, 0) and extremities of latus rectum are (3, 2) and (3, 2) is A) y2 = 2x - 4
15.
C) 0
The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2 = 4ax is another parabola with directrix A) x = -a
14.
B) 4
B) parabola with directix at x = 3/2 D) parabola with directix at x = -1/2
The equation of the parabola whose vertex and focus lie on the axis of x at distances a and a1 from the
origin respectively, is A) y2 = 4 (a1 - a) x 17.
B) y2 = 4 (a1 - a) (x - a) C) y2 = 4 (a1 - a) (x - a1) D) none of these
The vertex of a parabola is the point (a, b) and latus rectum is of length l. If the axis of the parabola is along the positive direction of y-axis, then its equation is A) (x + a)2 = l/2 (2y - 2b) C) (x + a)2 = l/4 (2y - 2b)
18.
The length of the latus rectum of the parabola 169 {(x - 1)2 + (y - 3)2 } = (5x - 12y + 17)2 is A) 12/13
19.
B) 14/13
95
C) a parabola
D) a hyperbola
B) parabola
C) hyperbola D) none
The angle made by a double ordinate of length 8a at the vertex of the parabola y2 = 4ax is A) /3
23.
B) a parabola with vertex at (2, 1) D) none of these
The curve represented by the equations x = sin2q, y = 2 cosq is A) ellipse
22.
D) none
The curve described parametrically by x = t2 + t + 1, y = t2 - t + 1 represents A) a pair of straight lines B) an ellipse
21.
C) 28/13
The parametric representation (2 + t2, 2t + 1) represents A) a parabola with focus at (2, 1) C) an ellipse with centre at (2, 1)
20.
B) (x - a)2 = l/2 (2y - 2b) D) (x-a)2 = l/8 (2y - 2b)
B) /2
C) /4
D) /6
If the segment intercepted by the parabola y2 = 4ax with the line lx + my + n = 0 subtends a right angle at the vertex, then
IIT- MATHEMATICS-SET-IV A) 4al + n = 0 24.
D) m = 3
B) x + a - b = 0
C) x + a + b = 0
D) x - a - b = 0
B) (-a/2, a/4)
C) (a/4, a/2)
D) (-a/4, a/2)
B) 600
C) 900
D) tan-1
B) (-2, 0)
C) (-1, 1)
D) (2, 0)
B) c = am + a/m
C) c = a + a/m
D) none of these
B) -1/2, 2
C) -2, 1/2
D) 1/2, 2
B) circle
C) parabola
D) ellipse
If the chord of contact of tangents from a point P to the parabola y2 = 4ax touches the parabola = 4by then the locus of P is A) circle
35.
C) m = 4
The locus of point from which the two tangents drawn to a parabola be such that slope of one is thrice of the other is A) line
34.
B) m = 2
The focal chord of y2 = 16x is tangent to (x - 6)2 + y2 = 2, then the possible values of the slope of this chord, are A) 1, -1
33.
D) none
If y = mx + c touches the parabola y2 = 4a (x + a), then A) c = a/m
32.
C) -1
y = x + 2 is any tangent to the parabola y2 = 8x. The point P on this tangent such that the other tangent from it which is perpendicular to it is A) (2, 4)
31.
B) 1
The portion of a tangent to a parabola cut off between the directirx and the curve subtends at the focus an angle A) 450
30.
D) (a, 3a)
If a tangent to the parabola y2=ax makes an angle 450 with x-axis, its point of contact will be A) (a/2, a/4)
29.
C) (-2a, 0)
Two straight lines are perpendicular to each other. One of them touches the parabola y2=4a (x+a) and the other touches y2=4b(x + b). Their point of intersection lies on the line A) x - a + b = 0
28.
B) (0, 2a)
The line y = mx + 1 is a tangent to the parabola y2 = 4x if A) m = 1
27.
D) al + n = 0
The straight line x + y = a touches the parabola y = x - x2 if a = A) 0
26.
C) 4am + n = 0
The straight line y = x + 2a touches the parabola y2 = 4a (x + a) at the point A) (-a, a)
25.
B) 4al + 4am + n = 0
B) parabola
C) ellipse
x2
D) hyperbola
Two tangents are drawn from the point (-2, -1) to the parabola y2 = 4x. If is the angle between these 96
PARABOLA tangents then tan = A) 1/2 36.
38.
D)
x y 1 1/3 0 1/3 b a (ab)2/3
If the line x + y = 1 touches the parabola y2 - y + x = 0, then the co-ordinates of the point of contact are B) (1/2, 1/2)
C) (0, 1)
D) (1, 0)
The point of intersection of the tangents at the ends of the latus rectum of the parabola y2 = 4x is B) (0, -1)
C) (-1, 0)
D) (1, 1)
Two tangents of the parabola y2 = 8x, meet the tangent at its vertex in the points P and Q. If PQ= 4, locus of the point of intersection of the two tangents is B) y2 = 8 (x - 2)
C) x2 = 8 (y - 2)
D) x2 = 8 (y + 2)
If the tangent at the point P (2, 4) to the parabola y2 = 8x meets the parabola y2 = 8x + 5 at Q and R then the mid-point of QR is B) (2, 4)
C) (7, 9)
D) none
The locus of the point of intersection of tangents to the parabola y2 = 4 (x + 1) and y2=8 (x+2) which are perpendicular to each other is A) x + 7 = 0
43.
D) none
x y 1 1/3 )=0 1/3 a b (ab) 2/3
C) xb1/3 + ya1/3 - (ab)2/3 = 0
A) (4, 2) 42.
C) m1m2 = 1
B
A) y2 = 8 (x + 2) 41.
B) m1m2 = -1
A) xa1/3 + yb1/3 + (ab)2/3 = 0
A) (-1, -1) 40.
D) 3
The equations of common tangent to the parabola y2 = 4ax and x2 = 4by is
A) (1, 1) 39.
C) 2
I f y + 3 = m1 (x + 2) and y + 3 = m2 (x + 2) are two tangents to the parabola y2 = 8x, then
A) m1 m2 = 0 37.
B) 1/3
B) x + 3 = 0
C) x - y - 4 = 0
D) x - y + 12 = 0
If y1 , y2 are the ordinates of two points P and Q on the parabola and y3 is the ordinate of the point of intersection of tangents at P and Q, then A) y1 , y2 , y3 are in A.P.. B) y1 , y3 , y2 are in A.P C) y1 , y2 , y3 are in G.P.. D) y1 , y3 , y2 are in G.P..
44.
Ordinates of three points A, B, C on the parabola y2 = 4ax are in G.P. Tangents at A and C intersect on A) line through B parallel to x-axis C) line through B and vertex of parabola
45.
If the tangents at P and Q on a parabola meet in T, then SP, ST and SQ are in A) A.P.
97
B) line through B || to y-axis D) line through B and focus of parabola
B) G.P.
C) H.P.
D) None of these
IIT- MATHEMATICS-SET-IV 46.
The tangent at P to a parabola meets the tangents at the vertex A in Q and S is the focus, then SP, SQ and SA are in A) A.P.
47.
bc 2
ab ba
2bc bc
D)
bc
B)
b ba
C)
a ba
D)
ab a b
B) axis
C) T.V.
D) none
B) a1/d2
C) 4a3/d2
D) d2/a
B) x + 2a = 0
C) x + 4a = 0
D) none
The angle between tangents to the parabola y2 = 4ax at the point where it intersects with the line x - y - a = 0 is A) /3
53.
C)
A chord of the parabola y2 = 4ax subtends a right angle at the vertex. The locus of the point of intersection of tangents at its extremities is A) x + a = 0
52.
bc bc
If a focal chord of the parabola be at a distance d from the vertex, then its length is equal to A) 2a2/d
51.
B)
If A1B1 and A2B2 are two focal chords of the parabola y2 = 4ax then the chords A1A2 and B2B2 intersect on A) directrix
50.
D) none
If b, c are the segments of a focal chord of the parabola y2 = 4ax, then c is equal to A)
49.
C) H.P.
If b and c are the lengths of the segments of any focal chord of a parabola y2 = 4ax, then the length of the semi-latus rectum is A)
48.
B) G.P.
B) /4
C) /6
D)/2
If P at12 , 2at1 and Q at22 , 2at2 are two variable points on the curve y2 = 4ax and PQ subtends a right angle the vertex, then is equal to A) -1
54.
D) -4
B) 1/4
C) 4
D) -4
If (2, -8) is at an end of a focal chord of the parabola y2 = 32x, then the other end of the chord is A) (-2, 8)
56.
C) -3
If the point P (4, -2) is the one end of the focal chord PQ of the parabola y2 = x, then the slope of the tangent at Q is A) -1/4
55.
B) -2
B) (32, -32)
C) (32, 32)
D) none
Angle between tangents drawn from the point (1, 4) to the parabola y2 = 4x is
98
PARABOLA A) /6 57.
C) /3
D) /2
Any tangent to a parabola y2 = 4ax and perpendicular to it from the focus meet on the line A) x = 0
58.
B) /4
B) y = 0
C) x = -a
D) y = -a
The two parabola y2 = 4x and x2 = 4y intersect at a point P, whose abscissa is a not zero, such that A) they touch each other at P B) they cut at right angles at P C) the tangents to each curve at P make complementary angles with the x-axis D) none of these
59.
Consider a circle with centre lying on the focus of the parabola y2 = 2px such that it touches the directrix of the parabola. Then a point of intersection of the circle and the parabola is A) (p/2, p)
60.
3 y = 3x + 1
D) none of these
B) 3 y = - (x + 3)
C)
3y=x+3
D)
3 y = - (3x + 1)
B) circle
C) parabola
D) hyperbola
B) y = 2x + 1
C) 2y = x + 8
D) y = x + 2
B) y = ± (x + 2a)
C) y = ± (2x + a)
D) none
B) 2
C) 1/2
D) none
If perpendicular be drawn from any two fixed points on the axis of a parabola at a distance d from the focus on any tangent to it, then the difference of their squares is A) a2 - d2
99
C) 15 5
The ratio of area of triangle inscribed in a parabola to the area of the triangle formed by the tangents at the vertices of the triangle is A) 1
67.
B) 10 5
Equations of the common tangents of the circles x2+y2 = 2a2 and the parabola y2 = 8ax are A) y = ± (x + a)
66.
D) x - 2y + 4 = 0
The equation of the common tangent to the curves y2 = 8x and xy = -1 is A) 3y = 9x + 2
65.
C) x - 2y - 4 = 0
Two parabolas y2 = 4a (x - ) and x2 = 4a (y - ) always touch each other, then the point of contact lies on (, , being parameters). A) straight line
64.
B) 2x + y - 4 = 0
The equation of the common tangent touching the circle (x-3)2+y2 = 9 and the parabola y2 = 4x above the x-axis is A)
63.
D) (-p/2, -p)
If y=2x+3 is a tangent to the parabola y2=24x, then its distance from the parallel normal is A) 5 5
62.
C) (-p/2, p)
The equation to the line touching both the parabolas y2 = 4x and x2 = -32y is A) x + 2y + 4 = 0
61.
B) (p/2, -p)
B) a2 + d2
C) 4ad
D) 2ad
IIT- MATHEMATICS-SET-IV 68.
The distance between a tangent to the parabola y2 = 4ax which is inclined to axis at an angle to axis and a parallel normal is A)
69.
a cos sin 2
B)
B) 3/4
B) y2 = 4a (x - a)
C) y2 = 2a (x + a)
D) none
C) - 9
D) - 3
B) c = 1/2
C) c > 1/2
D) none of these
B) 2
C) 3
D) 4
If two of the feet of normals drawn from a point to the parabola y2 = 4x be (1, 2) and (1, -2), then the third foot is
B) 2, 2 2
C) (0, 0)
D) none
The normal chord of the parabola y2 = 4ax at a point whose ordinate is equal to abscissa subtends a right angle at the A) focus
76.
D) 4
11 1 The number o distinct normals that can be drawn to the parabola y2 = 4x from the point is , 4 4
A) 2, 2 2 75.
a cos sin 2
Three normals to the parabola y2 = x are drawn through a point (c, 0), then
A) 1 74.
C) 2
B) 9
A) c = 1/4
73.
D)
If x + y = k is normal to y2 = 12x, then k is A) 3
72.
a sin cos 2
PNP’ is a double ordinate of the parabola y2 = 4ax then the normal at P and a line parallel to the axis through P’ meet on the parabola A) y2 = 4a (x - 2a)
71.
C)
If a normal chord of a parabola y2 = 4ax subtends a right angle at the vertex, then it is inclined at angle with the axis such that tan2 = A) 1/2
70.
a sin cos 2
B) vertex
C) ends of latus rectum D) none
If the normal to the parabola y2 = 4ax at the point P (at2, 2at) cuts the parabola again at Q (aT2, 2aT) then A) -2 T 2
77.
D) T2 8
If the normals at two points P and Q of a parabola y2 = 4ax intersect at a third point R on the curve, then the product of ordinates of P and Q is A) 4a2
78.
B) T (- - 8) (8, )C) T < 8
B) 2a2
C) - 4a2
D) 8a2
A triangle ABC of area D is inscribed in the parabola y2 = 4ax such that the vertex A lies at the vertex of 100
PARABOLA the parabola and BC is a focal chord. The differences of the distances of B and C from the axis of the parabola is B) 2/a2 D) none of these
A) 2/a C) a/2 79.
The locus of the middle points of the focal chord of the parabola y2 = 4ax is A) y2 = a (x - a)
80.
101
D) 2a
B) y2 = 2a (x - 4a)
C) y2 = a (x - 2a)
D) y2 = a (x + 2a)
B) G.P.
C) H.P.
D) none of these
B) y2 = a (x + a)
C) y2 = a (x - a)
D) none
B) y2 = 2 (x - 4)
C) y2 = 2 (x - 6)
D) none
B) ±
2
C) ±
3
D) none
A variable chord PQ of the parabola y2 = 4ax subtends a right angle at the vertex. The locus of the points of intersection of the normals at P and Q is the parabola A) y2 = 4a (x - 2a)
88.
C) a / 2
A is a point on the parabola y2 = 4ax. The normal at A cuts the parabola again at the point B If AB subtends a right angle at the vertex of the parabola, then the slope of AB is A) ± 1
87.
B) 2 2a
Through the vertex O of a parabola y2 = 4x chords OP and OQ are drawn at right angle to one-another. Then for all positions of P, PQ cuts the axis of the parabola at a fixed point and the locus of the middle point of PQ is A) y2 = 2 (x - 2)
86.
D) y = 0
The locus of the mid-points of the portion of the normal to the parabola y2 = 4ax intercepted between the curve and the axis is another parabola A) y2 = 2a (x - a)
85.
C) x = 2a
If the normals from any point to the parabola x2 = 4y cuts the line y = 2 in points whose abscissas are in A.P., then the slopes of the tangents at the three co-normal points are in A) A.P.
84.
B) x = -a
The locus of mid-points of the chords of the parabola y2 = 4ax which subtend a right angle at the vertex is A) y2 = 2a (x + 4a)
83.
D) none of these
The length of sub-normal to the parabola y2 = 4ax at any point is equal to A) a 2
82.
C) y2 = 4a (x - a)
The locus of the poles of focal chords of the parabola y2 = 4ax is A) x = 0
81.
B) y2 = 2a (x - a)
B) y2 = 16a (x - 6a)
C) y2 = 8a (x - 4a)
D) none
P, Q, R are the feet of the normals drawn to a parabola (y - 3)2 = 8 (x - 2). A circle cuts the above parabola in points P, Q, R and S. Then this circle always passes through the point.
IIT- MATHEMATICS-SET-IV A) (2, 3) 89.
B) y = -2x
C) y = –x/2
D) y = –2/3 x
B) (0, 16/3)
C) (0, 17/3)
D) (0, 18/3)
If the tangents at P and Q on a parabola meet in T, then SP, ST and SQ are in A) A. P.
92.
D) 2, 0)
A circle of radius 5 is dropped in the parabola x2 = 6y then the centre of the circle stands at A) (0, 15/3)
91.
C) (0, 3)
A parabola with focus at (1, 2) touching both the axes then the equation of its directrix is A) y = -x
90.
B) (3, 2)
B) G. P
C) H. P.
D) None of these
AB, AC are tangents to a parabola y 2 4ax, p1 , p2 , p3 are the lengths of the perpendiculars from A,B,C on any tangent to the curve, then p2, p1, p3 are in A) H. P.
93.
B) G. P
C) H. P
D) None of these
The normals at three points P, Q, R of the parabola y 2 4ax meet in (h,k). The centroid of triangle PQR lies on A) x = 0
94.
B) y = 0
C) x = –a
D) y = a
The condition that the parabolas y 2 4c x d and y 2 4ax have a common normal other t h e n x–axis (a>0, c>0) is A) 2a<2c+d
B) 2c<2a+d
C) 2d<2a+c
D) 2d<2c+a
95.
The vertex of the parabola is the point (a,B) and latus rectum is of length l. If the axis of the parabola is along the positive direction of y–axis then its equation is 1 1 2 2 A) x a y 2b B) x a y 2b 2 2 2 C) x a l y b D) None of these
96.
Let a be the angle which a tangent to the parabola makes with its axis, the distance tangent and parallel normal will be A) a sin 2 cos 2
97.
D) a cos 2
B) 4
C) 0
D) 8
If y+b=m1(x+a) and y+b=m2 (x+a) are two tangents to the parabola then A) m1 m2 0
99.
C) a tan 2
The equation ax 2 4 xy y 2 ax 3 y 2 0 represents a parabola if a is A) –4
98.
B) a cos ec sec2
between the
B) m1m2 1
C) m1m2 1
D) None of these
The triangle formed by the tangent to the parabola y x 2 at the point whose abscissa is x0 x0 1, 2 ,
102
PARABOLA the y–axis and the straight line y x02 has the greatest area if x0 A) 0
B) 1
C) 2
D) 3
100. Two parabolas C and D intersect at two different points, where C is y x 2 3 and D is y kx 2 . The intersection at which the x value is positive is designated point A, and x=a at this intersection, the tangent line l at A to the curve D intersects curve C at point B, other than A. If x–value of point B is 1 then a = A) 1
B) 2
C) 3
D) 4
101. If the normals from any point to the parabola x 2 4 y cuts the line y=2 in points whose abscissae are in are in A.P., then the slope of the tangents at the three co-normal points are in A) A.P
B) G.P
C) H.P.
D) None of these
102. P is a point which moves in the x–y plane such that the point P is nearer to the centre of a square than any of the sides. The four vertices of the square are a, a . The region in which P will move is bounded by parts of parabolas of which one has the equation A) y 2 a 2 2ax
B) x 2 a 2 2ay
C) y 2 2ax a 2
D) AN
103. The vertex of a parabola is (a,0) and the directrix is x+y=3a. The equation of the parabola is A) x 2 2 xy y 2 6ax 10ay 7a 2 0
B) x 2 2 xy y 2 6ax 10ay 7 a 2
C) x 2 2 xy y 2 6ax 10ay 7a 2
D) None of these
104. The equation of the parabola whose vertex and focus are on the positive side of the x– axis at d i s tance a and b respectively from the origin is A) y 2 4(b a)( x a ) B) y 2 4(a b)( x b) C) x 2 4(b a)( y a )
D) None of these
105. A ray of light moving parallel to the x–axis gets reflected from a parabolic mirror whoseequation is (y– 2)2 = 4(x+1). After reflection, the ray must pass through the point. A) (0,2)
B) (2,0)
C) (0,–2)
D) (–1,2)
106. If two tangents drawn from the point (,) to the parabola be such that the slope of one tangent is double of the other then 2 2 A) 9
B)
2 2 9
C) 2 9 2
D) None of these
107. The equation of the common tangent to the equal parabolas y 2 4ax and x 2 4ay is A) x+y+a=0
B) x+y=a
C) x–y=a
D) None of these
108. The set of point on the axis of the parabola y 2 4 x 8 from which the 3 normals to the rabola are all real and different is 103
p a -
IIT- MATHEMATICS-SET-IV A) {(k,0) : k 2 }
B) k , 0 : k 2
C)
0, k : k 2
D) None of these
109. The locus of the points of trisection of the double ordinates of the parabola y 2 4ax is A) y 2 ax
B) 9 y 2 4ax
C) 9 y 2 ax
D) y 2 9ax
110. If the normals at two points P and Q of a parabola y 2 4ax intersect at a third point R
on the
curve, then the product of ordinates of P and Q is A) 4a 2
B) 2a 2
C) 4a 2
D) 8a 2
111. If the line y 3x 3 0 cuts the parabola y 2 x 2 at A and B, then PA. PB is equal to [where P
A)
4
32
3, 0
3
B)
4 2 3
3
C)
4 3 2
D)
2
32
3
112. If tangents at A and B on the parabola y 2 4ax intersect at point C then ordinates of A, B
C and
are
A) Always in A.P
B) Always in G. P
C) Always in H. P
D) None of these
113. The set of points on the axis of the parabola y 2 4 x 2 y 5 0 from which all the three normals to the parabola are real is A) k , 0 ; k 1
B) k 1 ; k 3
C) k , 2 ; k 6
D) k ,3 ; k 8
114. The condition that the straight line lx+my+n=0 touches the parabola x 2 4ay is A) bn=am2
B) al2–mn=0
C) ln=am2
D) am=ln2
115. The equation to the line touching both the parabolas y 2 4 x and x 2 32 y is A) x 2 y 4 0
B) 2 x y 4 0
C) x 2 y 4 0
2
116. The length of the latus rectum of the parabola 169 x 1 y 3 A) 14/13
B) 12/13
C) 28/13
D) x 2 y 4 0 2
5x 12 y 17
2
is
D) None of these
117. If the normal at P (t) on y 2 4ax meet the curve again at Q, the point on the curve, the normal at which also passes through Q has co–ordinates (…….,…). 2a 2a A) 2 , t t
4a 2a B) 2 , t t
4a 4a C) 2 , t t
4a 8a D) 2 , t t
104
PARABOLA 118. Consider a circle with its centre lying on the focus of the parabola y 2 2 px such that it touches the directrix of the parabola. Then a point of intersection of the circle and the parabola is p A) , p 2
p B) , p 2
p C) , p 2
p D) , p 2
119. If the tangent at P on y 2 4ax meets the tangent at the vertex in Q, and S is the focus of the parabola, then SQP A) /3
B) /4
C) /2
D) 2/3
120. The diameter of the parabola y 2 6 x corresponding to the system of parallel chords 3x y c 0 , is
A) y 1 0
B) y 2 0
C) y 1 0
D) y 2 0
121. A line L passing through the focus of the parabola y 2 4 x 1 intersects the parabola in two distinct points. If ‘m’ be the slope of the line L then A) m(–1,1)
B) m(– , –1) 1, C) mR
D) None of these
122. Given the two ends of the latus rectum, the maximum number of parabolas that can be drawn is A) 1
B) 2
C) 0
D) infinite
123. If the vertex –(2,0) and the extremities of the latus rectum are (3,2) and (3,–2) then the equation of the parabola is A) y 2 2 x 4
B) x 2 4 y 8
C) y 2 4 x 8
D) None of these
124. The point (a, 2a) is an interior point of the region bounded by the parabola y 2 16 x and the double ordinate through the focus. Then a belongs to the open interval A) a<4
B) 0
C) 0
D) a>4
125. A chord PP of a parabola cuts the axis of the parabola at 0. The fet of the perpendiculars fr o m P and P on the axis are M and M respectively. If V is the vertex then VM, VO, VM are in A) AP
B) GP
C) HP
126. A focal chord of meets in P and Q. If S is the focus, then A) 1/a
B) 2/a
105
1 1 SP SQ
C) 4/a 2
127. The equation of a parabola is 25 x 2 y 5 A) vertex = (2,–5)
D) None of these
2
D) None of these
= 3x 4 y 1 . For this parabola
B) focus=(2,–5)
2
IIT- MATHEMATICS-SET-IV C) focus = (-2, 5)
D) axis has the equation 3x+4y–1=0
128. Let there be two parabolas with the same axis, focus of each being exterior to the other a n d the latus recta being 4a and 4b. The locus of the middle points of the intercepts between the parabolas made on the lines parallel to the common axis is a A) straight line if a=b C) parabola for all a,b
B) parabola if a = b D) None of these.
***
KEY 1.
2.
3.
4.
ac
d
a
a
16.
b
31.
b
46.
b
17. 18. 19.
b
c
b
32. 33. 34.
a
c
d
47. 48. 49.
c
a
a
5.
6.
abcd d
7.
8.
9.
10.
11.
abd
c
b
d
c
20. 21. 22. 23. 24. 25. 26.
c
b
b
a
b
b
a
35. 36. 37. 38. 39. 40. 41.
d
b
a
c
c
a
b
50. 51. 52. 53. 54. 55. 56.
c
c
d
d
c
c
c
12. 13. 14. 15.
b
c
b
c
27. 28. 29. 30.
c
c
c
b
42. 43. 44. 45.
b
b
b
b
57. 58. 59. 60.
a
c
ab
d
106
PARABOLA
61.
c
76.
d
91.
b
62. 63. 64.
c
d
d
77. 78. 79.
d
a
b
92. 93. 94.
b
b
a
65. 66. 67. 68. 69. 70. 71.
b
b
c
b
c
b
b
80. 81. 82. 83. 84. 85. 86.
b
d
b
b
c
a
b
72. 73. 74. 75.
c
b
c
a
87. 88. 89. 90.
b
a
b
d
95. 96. 97. 98. 99. 100. 101. 102.103. 104. 105.
b
b
b
c
c
c
b
abc b
b
a
106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117.118. 119. 120.
b
a
d
b
d
a
a
b
121. 122. 123. 124. 125. 126. 127. 128.
d
107
b
c
b
b
a
bc
ab
b
d
c
c
ab
c
a
IIT- MATHEMATICS-SET-IV
SECTION B - MORE ONE ANSWER TYPE QUESTIONS 1.
The parabola x2+2x-4y = 0 has A) vertex = (-1, -1)
2.
B) 81x-8y-162 = 0
C) 9x-4y+4=0
D) x-4y-36=0
C) 1
D) none of these
B) x1, x3, x2 are in GP
C) y1, y3, y2 are in AP
D) y1, y3, y2 are in GP
B) it is a parabola
C) its latus rectum = a
D) its latus rectum = 2a
A tangent to the parabola y2 = 4ax is inclined at p/3 with the axis of the parabola. The point of contact is a
2a 3
B) (3a, -2 3 a)
C) (3a, 2 3 a)
a 2a 3
D) 3 ,
A chord PP’ of a parabola cuts the axis of the parabola at O. The feet of the perpendiculars from P and P’ on the axis are M and M’ respectively. If V is the vertex then VM, VO, VM’ are in A) AP
10.
D) (-16, -8)
The equation of a locus is y2 + 2ax + 2by + c = 0. Then
A) 3 , 9.
C) (-16,8)
If the tangents to the parabola y2 = 4ax at (x1, y1), (x2, y2) cut at (x3, y3) then
A) it is an ellipse 8.
B) (16, -8)
B) -2
A) x1, x3, x2 are in AP 7.
B) focus == (2, -5) D) axis has the equation 3x+4y-1=0
If the tangents drawn from the point (0, 2) to the parabola y2 = 4ax are inclined at an angle 3/4 then the value of a is A) 2
6.
1 4
The equation of a tangent to the parabola y2 = 9x from the point (4, 10) is A) x - 4y + 36 = 0
5.
D) focus = 0,
Let PQ be a chord of the parabola y2 = 4x. A circle drawn with PQ as a diameter passes through the vertex V of the parabola. If ar (PVQ) = 20 unit2 then the coordinates of P are A) (16,8)
4.
3 4
The equation of a parabola is 25{(x-2)2 + (y+5)2} = (3x+4y-1)2. For this parabola A) vertex = (2, -5) C) directrix has the equation 3x + 4y-1= 0
3.
C) focus = 1,
B) latus rectum = 4
B) GP
C) HP
D) none of these
Let the equations of a circle and a parabola be x2 + y2 - 4x - 6 = 0 and y2 = 9x respectively. Then A) (1, -1) is a point on the common chord of contact B) the equation of the common chord is y+1=0 C) the length of the common chord is 6 D) none of these
11.
The equation of a common tangent to the parabola y2 = 2x and the circle x2+y2+4x=0 is A) 2 6x y 12
B) x+2 6 y+12 = 0
C) x-2 6 y+12=0
D) 2 6 x-y=12 108
PARABOLA 12. Let there be two parabolas with the same axis, focus of each being exterior to the other and the latus recta being 4a and 4b. The locus of the middle points of the intercepts between the parabolas made on the lines parallel to the common axis is a A) straight line if a = b 13.
17.
B) x2 = a2 + 2ay
C) y2 + 2ax = a2
D) none of these
B) y = e x / 2 a
C) y = ax
D) x2 = 4ax
Consider a circle with its centre lying on the focus of the parabola y2 = 2px such that it touches the directrix of the parabola. Then a point of intersection of the parabola. Then a point of intersection of the circle and the parabola is p A) , p 2
16.
D) none of these
Which one of the following curves cuts the parabola y2 = 4ax at right angles? A) x2 + y2 = a2
15.
C) parabola for all a,b
P is a point which moves in the x-y plane such that the point P is nearer to the centre of a square than any of the sides. The four vertices of the square are (±a, ±a). The region in which P will move is bounded by parts of parabolas of which one has the equation A) y2 = a2 + 2ax
14.
B) parabola if a b
p , p C) 2
p B) , p 2
p , p D) 2
Equation y2 - 2x - 2y + 5 = 0 represents: A) a pair of straight lines
B) a parabola with centre (1, 1)
C) a parabola with vertex (2, 1)
D) a parabola with directrix x = 3/2
A tangent to the parabola y2 = 8x makes an angle 450 with the line y = 3x + 5 then A) equation of tangent may be y + 2x -1 = 0 B) equation of tangent may be y + 2x + 1 = 0 C) equation of tangent may be 2y - x - 8 = 0 D) All above
18.
If from P(-1, 2) tangents PA and PB are drawn tot he parabola y2 = 4x then A) Equation of AB must be y = x - 1 C) Length of AB must be 4
19.
Equation of a tangent to the parabola y2 = 7x which is inclined at an angle of 450 to its axis A) 4x - 4y + 7 = 0
109
B) Length of AB must be 8 D) All above
B) 4x + 4y + 7 = 0
C) 7x - 7y + 4 = 0
D) 7x + 7y + 4 = 0
IIT- MATHEMATICS-SET-IV
KEY 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12. 13. 14. 15.
bc
bc
ab
ac
ab
bc
bd
ad
b
ac
bc
ab abc
16.
17. 18. 19.
cd
bc
ab
bd
ab
ab
110
PARABOLA
111
IIT-MATHEMATICS-SET-IV
5
ELLIPSE
112
ELLIPSE An ellipse is the locus of a point which moves in such a way that its distance from a fixed point is in a constant ratio (less than one) to its distance from a fixed line. The fixed point is called the focus and fixed line is called the directrix and the constant ratio is called the eccentricity of the ellipse.
EQUATION OF AN ELLIPSE Standard Equation of Ellipse Let S be the focus and ZM the directrix of the ellipse. Draw SZ ZM. Divide SZ internally and externally in the raio e : 1(e < 1) and let A and A be these internal and external points of division. Then SA = eAZ
... (1)
and SA = eAZ
... (2)
Clearly A and A will lie on the ellipse. Let AA = 2a and take C the mid point of AA as origin.
CA CA a ... (3)
Let P(x, y) by any point on the ellipse referred to CA and CB as co-ordinate axes. Then adding (1) and (2) we get
SA + SA e (AZ AZ)
Y (0, b) B
P(x, y)
Directrix
Directrix
AA e(CZ CA CA CZ) (from figure)
N
Z’ C A’ S’(-ae,0) S(ae,0) A x’ (–a, 0) (a, 0) (0, -b) B’ x = –a/e
Y’
M Z x Axis
x = a/e
AA e(2CZ) ( CA CA ) 2a = 2eCZ
CZ = a/e
The equation of the directrix MZ is x = CZ = a/e
Subtracting (1) from (2), we get SA SA e (AZ AZ)
(CA CS) (CA CS) e (AA) 2CS e (AA) 2CS = e(2a) 113
The focus S is (CS, 0) i.e., (ae, 0)
CS = ae
( CA CA)
IIT-MATHEMATICS-SET-IV Now draw PM MZ
SP e PM
or,
a (x – ae)2 + (y – 0)2 = e2 x e
(SP)2 = e2 (PM)2
2
(x – ae)2 + y2 = (a – ex)2
x2 + a2e2 – 2aex + y2 = a2 – 2aex + e2x2 x2 (1 – e2) + y2 = a2 (1 – e2)
x2 a
2
y2 2
2
a (1 e )
1
or
x2 a
2
y2 b
2
1 , where b2 = a2 (1 – e2)
This is the standard equation of an ellipse. AA and BB are called the major and minor axes of the ellipse. Here b < a and A and A are the vertices of the ellipse.
Another definition of Ellipse PS + PS = e (PM + PN) 2a e e
{ PM PN
2a } e
= 2a In other words an ellipse is locus of a point which moves in a plane so that the sum of its distances from fixed points is constant. Hence an ellipse can be also be defined as the locus of the point (P) which moves such that sum of its distances from two fixed points (S and S ) is a constant (2a).
Various terms related with an Ellipse Let the equation of the ellipse be
x2 y2 1, a b a2 b2
(1)
Vertices of an ellipse : The points at which the ellipse cuts the x-axis (a, 0) & (–a, 0) are called the vertices of the ellipse.
(2)
Major & Minor axis : The line segment AA is called the major axis and BB is called the minor axis. The major and minor axis taken together are called the principal axes and their lengths will be given by 2a and 2b respectively.
(3)
Centre: The point which bisect each chord of the ellipse is called the centre ((0, 0)).
(4)
a Directrix: ZM and ZM are two directrices and their equations are x and e x
(5)
a . e
Focus: S(–ae, 0) and S (ae, 0) are two foci of the ellipse.
114
ELLIPSE (6)
Latus Rectum: The chord which passes through either focus and perpendicular to the major axis is called a latus rectum.
Length of Latus Rectum : Length of Latus rectum is given by (7)
2b 2 . a
Relation between constant a, b and e b2 = a2 (1– e2) e2 =
a 2 b2 e= a2
a 2 b2 a
Result: x 2 y2 (1) Another form of standard equation of ellipse 2 2 1 when a < b a b In this case major axis is BB = 2b which is along y-axis and minor axis is AA = 2a which is along x-axis. Foci are S(0, be) and S (0, –be) and directrices are y = b/e and y = – b/e.
x 2 y2 (2) Focal distances: The focal distance of the point (x, y) on the ellipse 2 2 1 are a + a b ex and a – ex.
General equation of the ellipse The general equation of an ellipse, whose focus is (h, k) , the directrix is the line ax + by + c = 0 and the eccentricity is e is given by (x – h)2 + (y –k)2 =
e 2 (ax by c) 2 a 2 b2
.
Note: Condition for a second degree equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 in x & y to represent an ellipse is given by h2 – ab < 0 & = abc + 2 fgh – af 2 – bg2 – ch2 0
Parametric form of an ellipse Clearly x = a cos , y = bsin satisfy the equation
x 2 y2 1 for all real values of . a 2 b2
Moreover any point on the ellipse can be represented as (a cos , b sin ), 0 2 . Hence the parametric equations of the ellipse the parameter.
115
x 2 y2 1 are x = a cos y = bsin , where , is a 2 b2
IIT-MATHEMATICS-SET-IV y Q(acos, bsin)
A’
C
P
A
M
x
The point P (a cos , b sin ) is also called the point . The angle is called the eccentric angle of the point P on the ellipse. Draw a circle with AA (the major axis) as a diameter. The cirlce is called the auxiliary circle of the ellipse. The equation of the circle is x2 + y2 = a2. Draw QM as perpendicular to AA cutting the ellipse at P. The x-co-ordinate of P = CM = a cos y-coordinate of P is b sin P ( a cos , b sin ) Equation of the chord of the ellipse whose eccentric angles are & , x y cos sin cos a 2 b 2 2
POINT AND ELLIPSE x 2 y2 Let P (x1 y1) be any point and let 2 2 1 be the equation of an ellipse. a b The point lies outside, on or inside the ellipse according as S1 =
x12 y12 1 0 , = 0, < 0 a 2 b2
ELLIPSE AND LINE x 2 y2 Let the equation of an ellipse be 2 2 1 and the given line be y = mx + c. a b Solving the line and ellipse, we get
x 2 (mx c) 2 =1 a2 b2 i.e., (a2m2 + b2)x2 + 2 mca2 x + a2 (c2 – b2) = 0 above equation being a quadratic in x its discriminant = 4m2 c2a4 – 4a2 (a2 m2 + b2) (c2 – b2) 116
ELLIPSE = – b2 {c2 – (a2 m2 + b2)} = b2 {(a2m2 + b2) –c2} Hence the line intersects the ellipse in 2 distinct points if a2m2 + b2 > c2, in one point if c2 = a2m2 + b2 and does not intersect if a2m2 + b2 < c2.
y = mx
(a 2 m 2 b 2 ) touches the ellipse and condition for tangency is
c2 = a2m2 + b2 Moreover the line y = mx
(a 2 m 2 b 2 )
a 2m x 2 y2 1 at , touches the ellipse a 2 b 2 2 2 2 a m b
b2
. a m b 2
2
Note: (1)
x cos + y sin = p is a tangent if p2 = a2 cos2 + b2 sin2 .
(2)
lx + my + n = 0 is a tangent to the ellipse if n2 = a2 l2 + b2m2.
Equation of the Tangent (i)
x2 y2 xx yy The equation of the tangent at any point (x1, y1) on the ellipse 2 2 1 is 21 21 1 a b a b
(ii)
The equation of tangent at any point ‘ ’ is x y cos sin 1 a b
(iii)
Point of intersection of tangents to ellipse at points ‘ ’, ‘ ’ is
a cos b sin 2 , 2 cos cos 2 2
Equation of the Normal (i)
The equation of the normal at any point (x1, y1) on the ellipse a 2x b2y 2 a b2 x1 y1
(ii)
The equation of the normal at any point ‘ ’ is ax sec – by cosec = a2 – b2
117
x 2 y2 1 is a 2 b2
IIT-MATHEMATICS-SET-IV
EQUATION OF CHORD WITH MID POINT (x1, y1) x 2 y2 The equation of the chord of the ellipse 2 2 1 , whose mid point is (x1, y1) is given by T a b = S1, where T
xx1 yy1 x12 y12 1 S 1. and 1 a2 b2 a2 b2
CHORD OF CONTACT x 2 y2 If PA and PB be the tangents through point P(x1, y1) to the ellipse 2 2 1, a b then the equation of the chord of contact AB is
xx1 yy 21 1 or T = 0 at (x1, y1) 2 a b
Y A
P (x1, y1)
X
X
O B
Y
PAIR OF TANGENTS x 2 y2 Let P (x1, y1) be any point outside the ellipse 2 2 1, and let a pair of tangents PA, PB a b be drawn to it from P. then the equation of pair of tangents of PA and PB is SS1 = T2, where S
S1
x2 y2 1, a2 b2
x 2 y2 1 and a 2 b2 A
P(x , 1 y
1
)
B
T
xx1 yy1 2 1 a2 b
118
ELLIPSE
Note: x 2 y2 The locus of the point of intersection of the tangents to an ellipse 2 2 1 which are a b perpendicular to each other is called the director circle and its equation is given by x2 + y2 = a2 + b2.
Some Important Points to Remember (i)
The product of the lengths of the perpendicular segments from the foci on any tangent to the ellipse (at p) is b2 and the feet of these perpendiculars Y, Y lie on the auxiliary circle.
(ii)
If the normal at any point P on the ellipse with centre C meet the major & minor axes in G & g respectively, & if CF be perpendicular upon this normal, then (a) PF. PG = b2 (b) PF.Pg = a2
y x = a/e Y t Y 1
P Y’
C x’
S
G F N S’
T
x
g
y’
(c) PG. Pg = SP . S P (d) CG . CT = CS2 (e) locus of the mid point of Gg is another ellipse having the same eccentricity as that of the original ellipse. [S and S are the focii of the ellipse and T is the point where tangent at P meet the major axis].
119
(iii)
The tangent & normal at a point P on the ellipse bisect the external & internal angles between the focal distances of P. This refers to the well known reflection property of the ellipse which states that rays from one focus are reflected through other focus & vice-versa.
(iv)
The portion of the tangent to an ellipse between the point of contact & the directrix subtends a right angle at the corresponding focus.
(v)
The circle on any focal distance as diameter touches the auxiliary circle.
(vi)
If the tangent at the point P of a standard ellipse meets the axes in T and t and CY1 is the perpendicular on it from the centre then,
IIT-MATHEMATICS-SET-IV 2
2
(a) Tt. PY1 = a – b
(b) least value of Tt is a + b.
120
ELLIPSE
121
IIT-MATHEMATICS-SET-IV
5-A
ELLIPSE
122
ELLIPSE
WORKEDOUT ILLUATRATION ILLUSTRATION : 01 The set of values of a for which 13x 12 13y 22 a5x 12y 12 represents an ellipse, is A) 1 < a < 2
B) 0 < a< 1
C) 2 < a < 3
D) None of these
Solution : 2
2
We have 13 x 1 13 y 2 a 5 x 12 y 1
L.H.S. is +ve
R.H.S is also +ve then a > 0
2
…….(1)
……………(2)
Again equation (1) can be written as
169 25a x 2 169 144a y 2 120axy ... x ... y 5 a 0 It is an ellipse.
H 2 AB then 3600a 2 169 25a 169 144a a<1
………………..(3)
from (2) and (3) we get
0 a 1 ILLUSTRATION : 02
x2 y2 1 on the tangent at any point P,, a2 b2 and G is the point when the normal at P meets the major axis, then CF. PG = If CF is the perpendicular from the centre C of the ellipse
A) a 2 Solution :
123
B)ab
C) b 2
D) b3
IIT-MATHEMATICS-SET-IV Ellipse is
x2 y2 1 a2 b2
Tangent at ‘P’ is
CF
x y cos sin 1 a b
1 cos 2 sin 2 = 2 b2 a
ab
a
2
sin b 2 cos 2 2
a 2 b2 G cos , 0 Equation of normal at P = ax sec by cos ec a b then a 2
2
2
a 2 b2 2 PG a cos cos b sin 0 a b4 2 2 2 2 cos b sin a
=
=
b a
CF .PG b 2
a
2
sin b 2 cos 2
ILLUSTRATION : 03 The eccentricity of the ellipse ax 2 by 2 2 fx 2 gy c 0 if axis of ellipse parallel to x-axis is
ba A) b
B)
ab b
C)
ab a
D) None of these
Solution : ax 2 by 2 2 fx 2 gy c 0 2 fx 2 2 gy a x2 by c 0 a b 2
2
f g f 2 g2 a x b y c a b a b
124
ELLIPSE 2
2
f g x y a b 2 1 2 2 f f g g2 c c b b a a a b
if e eccentricity then f 2 g2 f 2 g2 c c b a b a 1 e2 b a
1 e2
a ba ba 2 e e b b b
ILLUSTRATION : 04 If A and B are two fixed points and P is a variable point such that PA + PB = 4, the locus of P is A) a parabola B) an ellipse
C) a hyperbola D) none of these
Solution :
PA PB 4 2
y2
2
y2 l & x a y2 m
x a
Let
x a
x a
2
y2 4
…………..(1)
2
l m 4ax, from (1)
…………..(2)
l m 4
l m
l m 4ax
l m ax adding (2) & (4) 2 l 4 ax
125
4l 16 a 2 x 2 8ax
………….(3) ………….(4) (from (2))
IIT-MATHEMATICS-SET-IV
4{x 2 y 2 2ax a 2 ] 16 a 2 x 2 8ax
4 a x 2
2
4 y 2 16 4 a 2 ellipse.
ILLUSTRATION : 05 The locus of the point of intersection of tangents to the ellipse
x y2 1 , which meet at right angles, a 2 b2
is A) a circle
B) a parabola
C) an ellipse
D) a hyperbola
Solution :
x y2 1 Equation of tangent in terms of slope a2 b2
y mx
a m 2
y mx
2
2
b2 a 2m2 b2
y 2 m 2 x 2 2mxy a 2 m 2 b 2 m 2 x 2 a 2 2mxy y 2 b 2 0
Which is perpendicular m1m2 1
y 2 b2 1 x 2 y 2 a 2 b 2 2 2 x a
ILLUSTRATION : 06 The distance of the point
A)
2
6 cos , 2 sin on the ellipse
B)
3 2
C)
5 2
x2 y2 1 from the centre is 2 if 6 2 D)
7 2
Solution : Since
6 cos , 2 sin lie on the ellipse | CP | 2 ;
x2 y 2 1 6 2
6 cos 2 2 sin 2 2
3 1 cos 2 1 cos 2 2
2cos2+2=0
cos 2 1
2 ,3 , 5 , 7
126
ELLIPSE
3 5 7 , , , 2 2 2 2
ILLUSTRATION : 07 If latus rectum of the ellipse x 2 tan 2 y 2 sec 2 1 is A) /12
B) /6
1 then is equal to 2
C) 5/12
D) None of these
Solution : x 2 tan 2 y 2 sec 2 1
x2 y2 1 cot 2 cos 2
cos 2 cot 2 1 e 2
sin 2 1 e2 , e 2 cos 2
e cos
90
latus rectum =
1 2b 2 2 a
0
a 4b 2
cot 4 cos 2
1 4 cos sin
sin 2
2 n 1
n
1 2
6
n n 1 for n = 1 2 12
5 2 12 12
ILLUSTRATION : 08 The parametric representation of a point on the ellipse whose foci are (-1, 0) and (7, 0) and eccentricity is
1 , is 2
A) 3 8cos , 4 3 sin C) 3 4 3 cos ,8sin Solution :
127
B) 8cos , 4 3 sin D)None of these
IIT-MATHEMATICS-SET-IV Foci are (-1, 0) & (7, 0) Co-ordinate of the centre is (3, 0)
1 7
Distance between foci =
2
0 8
2ae = 8
ae 4 Here e
1 2
ae 4 1 b 2 a 2 1 e 2 64 1 4
= 64 16 48 Equation of Ellipse with centre (3, 0) is
Parametric co-ordinates 3 8cos , 4 3 sin
ILLUSTRATION : 09 An ellipse has OB as semi-minor axis. F and F’ are its foci and the angle FBF’ is a right angle. Then eccentricity of the ellipse is A)
1 3
B)
1 2
C)
1 2
D)None of these
Solution : Let the ellipse be
x2 y2 1 . Then co-ordinates of F, B & F’ are (ae, 0), (0, b) (-ae, 0) respectively.. a 2 b2
The slope of BF =
& slope of BF '
b0 b 0 ae ae 0b b ae 0 ae
FBF ' 900
Slope of BF x slope of BF’ = -1
a 2 1 e 2 a 2 e 2
b2 1 a 2e2
b 2 a 2e2
128
ELLIPSE
e
1 2 Y B
900 F' O
F
X
***
129
IIT-MATHEMATICS-SET-IV
SECTION A - SINGLE ANSWER TYPE QUESTIONS 1.
S and S’ are the foci of an ellipse and B is an end of the minor axis. If SS’B is an equilateral triangle, the eccentricity of the ellipse is A) 1/4
2.
3 /2
386 38
2 /3
C)
3
D) none of these
B) 6,
5
C) 4, 2 5
D) 6+2 5 , 2 5
386 12
B)
C)
386 13
D)
386 25
x 2 y2 1 whose foci are S and , then PS + PS’ = 16 25
B) 8
C) 10
D) 12
B) 1/2 abe
C) 2abe
D) none
An ellipse has OB as a semi minor axis. F, F’ are its foci, and the angle FBF’ is a right angle. Then the eccentricity of the ellipse is A) 1/ 3
9.
D)
x2 y2 L et P be a variable point on the ellipse 2 2 1 with foci S1 and S2. If A be the area of the triangle a b PS1S2, then the maximum value of A is A) abe
8.
B) 2/ 3
If P is a point on the ellipse A) 6
7.
C) 1 / 3
If (5, 12) and (24, 7) are the foci of a conic passing through the origin, then the eccentricity of conic is A)
6.
B) 1/3
An ellipse is described by using an endless string which is passed over two pins. If the axes are 6cm and 4cm, the necessary length of the string and the distance between the pins respectively in cms are A) 6, 2 5
5.
D) 2/3
The eccentricity of the conic x2 - 4x + 4y + 4y2 = 12 is A)
4.
C) 1/2
The eccentricity of an ellipse whose pair of conjugate diameters are 2y = x and 3y = -2x is A) 2/3
3.
B) 1/3
B) 1/ 2
C) 1/2
D) none
A man running round a race course notes that the sum of the distances of two flag-posts from him is always 10 metres and the distance between the flag posts is 8 metres. The area of the path he encloses in square metres is A) 15
B) 12
C) 18
D) 8
130
ELLIPSE 10.
The eccentric angle of a point on the ellipse
x2 y2 1 whose distance from the centre of ellipse is 2 6 2
is A) 2100
11.
B) 2700
B) 3
The equation
B) parabola
C) hyperbola
D) circle
B) a circle
C) a pair of lines
D) a line
x2 y2 1 0 represents an ellipse if 2a a5 B) a > 5
C) 2 < a < 5
D) none of these
B) [4, 5]
C) (-, 0)
D) (0, )
( x y 2)2 ( x y )2 The centre of the ellipse = 1 is 9 16 A) (0, 0)
131
D) none
x2 y2 2 2 y = 1 meets the ellipse x 2 = 1 in four distinct points and a = b2 - 5b + 7 then If the ellipse 4 a b does not belong to A) [2, 3]
18.
C) an ellipse
The equation ( x 3) 2 ( y 1) 2 ( x 3) 2 ( y 1) 2 6 represents
A) a > 2
17.
B) a parabola
The curve represented by x = 3 (cos t + sin t), y = 4 (cost - sin t) is
A) an ellipse
16.
B) Q lies outside both C and E D) P lies inside C but outside E
A bar of given length moves with its extremities on two fixed straight lines at right angles. Any point of the rod describes
A) ellipse 15.
D) 7/2
x2 y2 1 and C be the circle x2 + y2 = 9. Let P and Q be the points (1, 2) and 9 4 (2, 1) respectively. Then
A) a circle 14.
C) 12
Let E be the ellipse
A) Q lies inside C but outside E C) P lies inside both C and E 13.
D) 450
x2 y 2 1 , and having its centre (0, 3) The radius of the circle passing through the foci of the ellipse 16 9 is A) 4
12.
C) 3000
B) (1, 1)
C) (1, 0)
D) (0, 1)
IIT-MATHEMATICS-SET-IV 19.
The equation (5x - 1)2 + (5y - 2)2 = (l2 - 2l + 1) (3x + 4l - 1)2 represents an ellipse if l Î A) (0, 1)
20.
x2 y 2 1 16 15
D) ± 8
e) ± (132)
B) 27/2
C) 27/4
D) 27/55
B)
a 2 b2
C) (a2 + b2)
D) none
B)
x2 y2 1 80 5 / 4
C)
x2 y 2 1 20 5
D)
x2 y2 1 5 16
x2 y 2 1 intersects the major axis and minor axis in points A 18 32 and B respectively, then area of DOAB in square units is If a tangent of slope -4/3 to the ellipse
A) 12 25.
C) ± 1
An ellipse passes through the point (4, -1) and touches the line x + 4y - 10 = 0. If its axes coincide with co-ordinate axes, then its equation is A)
24.
B) ± 6
x2 y2 If any tangent to the ellipse 2 2 1 intercepts equal length l on axes, then l is equal to a b A) a2 + b2
23.
D) (-1, 0)
x2 y2 1 at end of latus rectum. Find the area of quadrilateral so Tangents are drawn to the ellipse 9 5 formed A) 27
22.
C) (1, 2)
If the straight line y=4x+c is a tangent to the ellipse x2/8+y2/4 = 1, then c will be equal to A) ± 4
21.
B) (0, 2)
B) 24
C) 48
D) 64
The sum of the squares of the perpendiculars on any tangent to the ellipse x2/a2 + y2/b2 = 1 from two points the minor axis each distance a 2 b 2 from the centre is A) a2
26.
D) 2b2
B) 3
C) 4
D) 5
x y x2 y2 2 If touches the ellipse 2 2 1 at a point P, then eccentric angle of P is a b a b
A) 0 28.
C) 2a2
x2 y2 1 on the If F1 and F2 be the feet of the perpendiculars from the foci S1 and S2 of an ellipse 5 3 tangent at any point P on the ellipse then (S1F1) (S2F2) is equal to A) 2
27.
B) b2
B) 450
C) 600
D) 900
An ellipse slides between two perpendicular straight lines. Then the locus of its centre is 132
ELLIPSE A) circle 29.
C) ellipse
D) hyperbola
The line 2x + y = 3 cuts the ellipse 4x2 + y2 = 5 at P and Q. If be the angle between the normals at these points, then tan q = A) 1/2
30.
B) parabola
B) 3/4
C) 3/5
If the normal at one end of the latus rectum of an ellipse
D) 5
x2 y2 1 passes through the one end of the a 2 b2
minor axis, then A) e4 - e2 + 1 = 0
31.
1 1 2 1 2 x 2y
133
D) a2 - b2
B) -2/3
C) 3/2
D) -3/2
a 2 b2 B) 2 2 4 x y
C)
x2 y 2 4 a2 b2
D) none of these
B)
1 1 2 1 2 4x 2 y
C)
1 1 2 1 2 2x 4 y
D)
1 1 2 1 2 2x y
The locus of the point of intersection of tangents to an ellipse at two points sum of whose eccentric angles is constant is A) straight line B) circle
36.
C) a2 + b2
Tangents are drawn to x2 + 2y2 = 2. the locus of mid-point of intercept made by tangents between the axes is A)
35.
a 2 b2 B) a 2 b 2 ab
x2 y2 The locus of the mid-points of the portion of the tangents to the ellipse 2 2 1 intercepted bea b tween the axes is x2 y 2 A) 2 2 4 a b
34.
D) e4 + e2 - 1 = 0
x2 y2 1 intersects it again at the point Q (2), then cos If the normal at the point P (q) to the ellipse 14 5 is equal to A) 2/3
33.
C) e2 + e + 1 = 0
x2 y2 The area of rectangle formed by perpendiculars from the centre of ellipse 2 2 1 to the tangent a b and normal at the point whose eccentric angle is p/4 is
a 2 b2 A) a 2 b 2 ab
32.
B) e2 - e + 1 = 0
C) parabola
D) ellipse
x2 y2 The eccentric angles of extremities of a chord of an ellipse 2 2 1 are 1 and 2 . If this chord a b passes through the focus, then
IIT-MATHEMATICS-SET-IV A) tan
1 1 e . tan 2 0 2 2 1 e
B) cos
sin 1 sin 2 C) e sin( ) 1 2
37.
x 2 y 2 ex a 2 b2 a
B)
x 2 y 2 ex a 2 b2 a
2 1 D) , 5 5
D) none
B) a circle
C) a parabola
D) an ellipse
B) a circle
C) a parabola
D) a hyperbola
x2 y 2 1 in the first quadrant is The slope of the common tangent to the curves x + y = 16 and 25 4 1 3
B) -
1 3
2
C)
2 3
D) -
2 3
( x 3) 2 ( y 2) 2 The foci of the ellipse = 1 are 36 16
B) 3 2 5, 2
C) (3, -2)
D) none
The equation of the ellipse whose vertices are (-4,1), (6, 1) and a focal chord is x–2y–2=0 is A) (x – 1)2/25 + (y – 1)2/16 = 1 C) (x – 1)2/16 + (y – 1)2/25 = 1
45.
C) x2 + y2 = a2 + b2
2
A) 3 2 5, 2 44.
2 1 C) , 5 5
A variable ellipse has one focus fixed and it always touches two fixed intersecting lines. Then the locus of its centre is
A)
43.
D) (1 + sin2 )-2
An ellipse slides between two perpendicular lines then the locus of its centre is
A) a straight line
42.
C) (1+sin2 )-3/2
2 1 B) , 5 5
A) a straight line 41.
B) (1 + sin2 )-1/2
x2 y2 The locus of the mid-points of the focal chords of the ellipse 2 2 1 is a b A)
40.
1 e 1 . cot 2 2 2 e 1
On the ellipse 4x2 + 9y2 = 1, the points at which the tangents are parallel to the line 8x = 9y are 2 1 A) , 5 5
39.
D) cot
The tangent at a point P (a cos, b sin) of an ellipse x2/a2 + y2/b2 = 1, meets its auxiliary circle in two points, the chord joining which subtends a right angle at the centre, then the eccentricity of the ellipse is A) (1 + sin2 )-1
38.
1 2 e.cos 1 2 2 2
B) (x + 1)2/25 + (y + 1)2/16 = 1 D) (x + 1)2/16 + (y + 1)2/25 = 1
The focus is (2, 0). The directrix is x–y=0 and the eccentricity is 1/3. The equation o the ellipse is 134
ELLIPSE A) 18 (x2 + y2 – 4x + 4) = (x – y)2 C) 6[(x-2)2 + y2] = (x-y)2
46.
B) 9 (x2 + y2 – 4x + 4) = (x – y)2 D) non
If the chord joining the points a and b on the ellipse
x2 y 2 = 1 subtends a right angle at (, 0), then a 2 b2
tan /2, tan /2 = A) a2 / b2 47.
135
D) 4
B) 36/5
C) 4
D) 5
B) x2 + y2 = b2
C) x2- a2 = 2xy
D) (x2 – a2) = 2xy
B) 2b2
C) a2
D) 2a2
B) 4/13
C) 5/13
D) none
B) 1/ 1 sin 2
C) 1 cos 2
D) 1 sin 2
The tangent and the normal at P() on the ellipse x2 + 4y2 = 4 meet the major axis in Q and R respectively. If QR = 2, then cosq = A) 1/3
55.
C) 3
The tangent at a on the ellipse b2x2 + a2y2 = a2b2 meets the auxilary at two points P,Q. If PQ subtends a right angle at the centre of the ellipse then e = A) 1/ 1 cos 2
54.
B) 2
S(3, 4) and S1(9, 12) are the focii of an ellipse and the foot of the perpendicular from S to a tangent to the ellipse in (1, -4). Then the eccentricity of the ellipse is A) 3/13
53.
D) be a 2 x12
The tangent at P on the ellipse meets the minor axis in Q, and PR is drwn perpendicular to the minor axis and C is the centre. Then CQ.CR = A) b2
52.
C) ae b 2 x12
Tangents to the ellipse b2x2 + a2y2 = a2b2 make angles a and b with the major axis such that tan + tan = . The locus of their point of intersection is A) x2 + y2 = a2
51.
B) be b 2 x12
If PSP1 is a focal chord of the ellipse 16x2 + 25y2 = 400, and SP = 16, then S1P1 = A) 26/5
50.
D) –b2/a2
PSP1 is a focal chord of the ellipse 16x2 + 25y2 = 400. If SP = 8 then SP1 = A) 1
49.
C) b2/a2
If P(x1, y1) is a point on the ellipse b2x2 + a2y2 = a2b2 then the area of DSPS1 = A) ae a 2 x12
48.
B) –a2/b2
B) 2/3
C) 1
x2 y2 2 2 The locus of poles of normal chords of the ellipse 2 2 a b a b x2 y2 x2 y2 2 2 2 2 A) 4 4 a b B) 4 4 a b a b a b
D) none
a 6 b6 2 (a 2 b 2 )2 C) 2 x y 56.
a 4 b4 2 2 2 D) 2 2 (a b ) x y
IIT-MATHEMATICS-SET-IV
If the variable lines l1(x – a) + y = 0 and l2(x + a) + y = 0 are conjugate lines w.r. to the ellipse
x2 y2 1 . Then the locus of their point of intersection is a2 b2
A) 2
57.
x2 y 2 1 a 2 b2
B)
x2 y2 2 1 a2 b2
C)
x2 y 2 2 a 2 b2
The ratio of the ordinates of a point and its corresponding point is
D) none
2 2 then eccentricity of the ellipse 3
is A) 1/3
58.
B) 2/3
C)
2 /3
D) 2 2 / 3
x2 y 2 The points P lies on the ellipse = 1 whose focii are S and S1. The eq. of the external 4 4 2
angular bisector of SPS 1 of is A) x 2 y 2 2
59
B) 2 x 3 3 y 12
C) 3 x 4 y 12 2 0 D) x y 12 2 = 0
If and be the feet of the perpendicular from the foci and of an ellipse
x2 y 2 1 on the tangent at 5 3
any point P on the ellipse, then S1F1 S 2 F2 is equal to A) 2
60.
1
r 2 b2 a2 r 2
r 2 b2 B) 2 2 a r
r 2 b2 C) a 2 r 2
D)
a2 r 2 r 2 b2
B) 2/3
C) 3/4
D) none of these
x2 y2 1 represents an ellipse if The equation 10 a 4 a A) a < 4
63.
D) 5
The eccentricity of the ellipse 9 x 2 5 y 2 30 y 0 is A) 1/3
62.
C) 4
x2 y2 The slope of a common tangent to the ellipse 2 2 1 and a concentric circle of radius r is a b A) tan
61.
B) 3
B) a > 4
C) 4 < a < 10
If and are eccentric angles of the ends of a focal chord of the ellipse
D) a > 10
x2 y2 2 1 , then tan tan 2 2 2 a b
is equal to 136
ELLIPSE A)
64.
1 e 1 e
The centre of the ellipse A) (0,0)
67.
a 2 b2 B) 2 2 4 x y
x2 y 2 C) 2 2 4 a b
D) none of these
B)–5/6
x y 2 9
C) –2/3 2
x y 16
B) (1,1)
D) none of these
2
1 is C) (1,0)
D) (0,1)
B)6, 5
C)4,2 5
D) 6 + 2 5
B) 12sq.units
C) 36sq.units
B) e 2 e 1 0
If the normal at the point P to the ellipse
C) e 2 e 1 0
71.
x2 y 2 1 intersects it again at the point / 2 14 5
C) 3/2
D) -3/2
The eccentricity of an ellipse whose pair of a conjugate diameter are y = x and 3y = -2x is A) 2/3
137
B) -2/3
B) 1/3
end of
D) e 4 e 2 1 0
cos , then is equal to A) 2/3
t ri-
D) none of these
x2 y2 If the normal at one end of the latusrectum of an ellipse 2 2 1 passes through the one a b the minor axis, then A) e 4 e 2 1 0
70.
D) none of these
x2 y 2 1 with foci at S and S’. If A be the area of Let P be a variable point on the ellipse 25 16 angle PSS’, then the maximum value of A is A) 24sq.units
69.
e 1 e 1
An ellipse is described by using an endless string which is passed over two pins. If the axes are 6cm and 4cm the necessary length of the string and the distance between the pins respectively in cms. are A) 6, 2 5
68.
C)
AB is a diameter of x 2 9 y 2 25 . The eccentric angle of A is /6 Then the eccentric angle of B is A) 5/6
66.
e 1 e 1
x2 y2 The locus of the middle point of the portion of a tangent to the ellipse 2 2 1 included between the a b axes is the curve x2 y 2 A) 2 2 4 a b
65.
B)
C) 1/ 3
D) none of these
IIT-MATHEMATICS-SET-IV 72.
The locus of mid-points of a focal chord of the ellipse
x 2 y 2 ex A) 2 2 a b a
73.
x 2 y 2 ex B) 2 2 a b a
x2 y2 1 is a2 b2
C) x 2 y 2 a 2 b 2
The radius of the circle passing through the foci of the ellipse
D) None of these
x2 y 2 1 , and having its centre (0,3) 16 9
is A) 4 74.
B) 3
B) 1/3
C) 1/2
t ri-
D) 2/3
On the ellipse 4 x 2 9 y 2 1 the points at which the tangents are parallel to the line are 2 1 A) , 5 5
76.
D) 7/2
S and T are the foci of an ellipse and B is an end of the minor axis. If STB is an equilateral angle, the eccentricity of the ellipse is A) 1/4
75.
C) 12
2 1 B) , 5 5
2 1 C) , 5 5
2 3 D) , 5 5
The equation of the ellipse whose focus is 1, 1 directrix the line x y 3 0 and eccentricity 1/ 2 is
77.
A) 7 x 2 2 xy 7 y 2 10 x 10 y 7 0
B) 7 x 2 2 xy 7 y 2 7 0
C) 7 x 2 2 xy 7 y 2 10 x 10 y 7 0
D) none of these
x2 y2 The eccentricity of an ellipse 2 2 1 whose latus rectum is half of its major axis is a b
A) 78.
1 2
B)
2 3
C)
3 2
D) none of these
The equations of the tangents to the ellipse 4 x 2 3 y 2 5 , which are inclined at 600 tothe axis of x are (A) y 3x
65 12
B) y 3x
12 65
C) y
x 65 12 3
D) None of these
138
ELLIPSE
KEY
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
C
D
A
D
AB
C
A
B
A
A
A
16.
C
31.
B
17. 18. 19.
A
B
B
32. 33. 34.
B
A
C
20. 21. 22. 23. 24. 25. 26.
C
A
B
BC
B
C
B
35. 36. 37. 38. 39. 40. 41.
A ABCD B
12. 13. 14. 15.
D
C
A
D
27. 28. 29. 30.
B
A
C
D
42. 43. 44. 45.
BD
B
B
A
D
52 53
54
55
56
3
2
3
2
1
1
57 58
59
60
2
1
B
B
aabcdbc 46
47
48 49
50
51
4
4
2
4
1
61
62
63 64
65
66
67 68
69
70
71
72 73
74
75
B
A
B
B
B
D
D
A
C
A
C
BD
76
77
78
B
A
A
139
4
B
2
B
1
A
IIT-MATHEMATICS-SET-IV
SECTION B - MORE THAN ONE ANSWER TYPE QUESTIONS 1.
A focus of the hyperbola 25x2 - 36y2 = 225 is A)
2.
61,0
1 61,0 2
3
, 2 2
3 2
D)
, 2 2
3
C) (-5, 2)
D) (-5, -2)
B) y1 + y2 + y3 + y4 = 0 D) y1 y2 y3 y4 = c4
On the ellipse 4x2 + 9y2 = 1, the points at which the tangents are parallel to 8x = 9y are 2 1
A) 5 , 5
2 1
2 1
B) 5 , 5
C) 5 , 5
2
1
D) 5 5
Let L be the line 2x + y = 2. If the axes are rotated by 450 without transforming the origin, the intercepts made by the line L on the new axes are respectively A) 1 and 2
B)
2 2 and 2 3
C)
2 and 1
D) 2
2 and
2 2 3
The focal distances of the point (4 3 , 5) on the ellipse 25x2 + 16y2 = 1600 may be A) 7
8.
1 61,0 2
D)
The circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points, P(x1, y1), Q(x2, y2) R(x3, y3), S(x4, y4), then C) x1 x2 x3 x4 = c
7.
C) , 2
B) (5, 2)
4
6.
A point on the ellipse x2 + 3y2 = 37, where the normal is parallel to the line 6x - 5y = 2, is
A) x1 + x2 x3 + x4 = 0
5.
3 2
B) ,2
A) (5, -2) 4.
C) 61,0
The point P on the ellipse 4x2 + 9x2 = 36 is such that the area of the PF1F2 = 10 where F1, F2 are foci. Then P has the coordinates A)
3.
B)
B) 6
C) 13
D) 11
x2 y 2 If PSQ and PS’R be two focal chords, an ellipse 2 2 = 1 and P, Q, R are points a b (a cos , b sin ) (a cos 1, bsin 1) (a cos 2, b sin 2) respectively then we must have
1 e 1 A) tan tan 2 e 1 2
1 e 2 1 B) tan tan 2 2 e 1 2
2 1 e C) tan tan 2 1 e 2
2 1 e2 D) tan tan 2 1 e2 2
140
ELLIPSE 9.
x2 y 2 = 1 meets the circle x2 + y2 = a2 in Q. If P is (a cos , b sin ) and a2 b2 O is the centre of the ellipse then An ordinate NP of an ellipse
A) PON = C) normal at Q must be y = x tan
10.
x2 y 2 A tangent to the ellipse = 1 is cut by the tangent at the extrimities of the major axis at T and T’ 9 4 of the circle on TT’ as diameter passes through the point then Q may be A) (-
11.
B) QON = D) normal at P must be y = x tan
5 , 0)
B( 5 , 0)
C) (0, 0)
D) (3,2)
If m is variable then the locus of point of intersection of lines (ab 0)
x y x y 1 = m and must a b a b m
be A) parabola
12.
B) a hyperbola
C)
x2 y 2 =1 a2 b2
D) All above
x2 y 2 y 2 x2 Let 2 = 1and 2 2 = 1 be two hyperbola then the slope of their common tangent may be a b2 a b A) 1
B) 2
C) -
2
7.
8.
D) - 1
KEY
1.
BD
141
2.
3.
4.
AD BD ABCD
5.
6.
9.
10.
11. 12.
BD BD AC AC BC
AB
BC AD
IIT-MATHEMATICS-SET-IV
6
HYPERBOLA
142
HYPERBOLA The hyperbola is the locus of a point which moves such that its distance from a fixed point called focus is always e times (e > 1) its distance from a fixed line called directrix.
VARIOUS FORMS OF HYPERBOLA Standard form Let S be the focus and ZM the directrix of a hyperbola. Since e > 1, we can divide SZ internally and externally in the ratio e : 1, let the points of division be A and A as in the figure. Let AA = 2a and is bisected at C. Then, SA = e. AZ, SA = e. ZA SA + SA e (AZ ZA) 2ae i.e., 2SC = 2ae or SC = ae Similarly by subtraction, SA SA = e(ZA ZA) 2e.ZC 2a 2eZC ZC a / e. y M’
B
P
M
x
S’ A’
Z’
C
Z
A
S
B’
Now, take C as the origin, CA as the x-axis, and the perpendicular line CY as the y-axis. Then, S is the point (ae, 0) and ZM the line x = a/e. Let P(x, y) be any point on the hyperbola. Then the condition PS2 = e2. (distance of P from ZM)2 gives (x – ae)2 + y2 = e2 (x – a/e)2 or
2
2
2
2
2
x (1 – e ) + y = a (1 – e ) i.e.,
x2 y2 1 a 2 a 2 (e2 1)
... (i)
x 2 y2 Since e > 1, e – 1 is positive. Let a (e – 1) = b . Then the equation (i) becomes 2 2 1. a b 2
2
2
2
b2 x 2 y2 2 e 1 2 . The eccentricity e of the hyperbola 2 2 1 is given by the relation a b a Since the curve is symmetrical about the y-axis, it is clear that there exists another focus S at (–ae, 0) and a corresponding directrix ZM with the equation x = –a/e, such that the same hyperbola is described if a point moves so that its distance from S is e times its distance from ZM . (a) 143
Foci: S = (ae, 0) & S (–ae, 0)
IIT-MATHEMATICS-SET-IV a a & x e e
(b)
Equation of directrices: x =
(c)
Vertices: A = (a, 0) & A = (–a, 0)
(d)
Transverse Axis: The lines segment AA of length 2a in which the foci S & S both called Transverse axis of the Hyperbola.
(e)
Conjugate Axis: The line segment BB ( B (0, b)) and ( B (0, –b)) is called the Conjugate axis of the hyperbola. The Transverse axis & the Conjugate axis of the hyperbola are together called principal axes of the hyperbola.
(f)
(h)
b2 2b 2 b2 Length of latus rectum = a & L ae, a , L ae, a x 2 y2 Focal distance: The focal distance of any point (x, y) on the hyperbola x 2 2 1 are ex– a b a and ex + a
Another Definition of the Hyperbola The difference of the focal distances of a point on the hyperbola is constant. PM and PM are perpendiculars to the directrices MZ and M Z . PS PS e (PM PM) . y M’
M
P x
S’
Z’
C
Z
S
eMM e(2a / e) 2a = constant.
Parametric Coordinates x 2 y2 We can express the coordinates of a point P(x, y) of the hyperbola 2 2 1 in terms of a a b single parameter, say . In the adjacent figure OM = a sec and PM = b tan . Thus any point on the curve, in parametric form is x = a sec , y b tan In other words, (a sec , b tan ) is a point on the hyperbola for all values of 144
HYPERBOLA (2n 1) 2 , n I
The point (a sec , b tan ) is briefly called the point . P(x, y) N a
O
x M
x 2 + y 2 = a2
Note: The circle having equation x2 + y2 = a2 is called the auxilliary circle of the hyperbola
General Form The equation of hyperbola, whose focus is point (h, k), directrix is lx + my + n = 0
e 2 (lx my n ) 2 & ecentricity ‘e’ is given by (x – h) + (y – k) = (l 2 m 2 ) 2
2
Conjugate Hyperbola The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called the conjugate hyperbola of the given hyperbola.
x 2 y2 x 2 y2 e.g., 2 2 1 & 2 2 1 a b a b are conjugate hyperbolas of each.
Note: If e1 & e2 are the eccentricities of the hyperbola & its conjugate then
1 1 1. e12 e22
ASYMPTOTES Definition: If the length of perpendicular drawn from a point on the hyperbola to a straight line tends to zero as the point on moves to infinity. The straight line is called asymptotes.
x 2 y2 Let y = mx + c is the asymptote of the hyperbola 2 2 1 . Solving these two we get the a b 2 2 2 2 2 2 2 2 quadratic as (b – a m )x – 2a mcx–a (b + c ) = 0 In order that y = mx + c be an asymptote, both roots of equation (1) must approach infinity. which are: coefficient of x2 = 0 & coefficient of x = 0
145
IIT-MATHEMATICS-SET-IV or
m=
b & a2mc = 0 c = 0 a
... (1)
Y BP
Q
A
A C
X
R B S
equation of asymptote are
x y x y 0& 0. a b a b
Obviously angle between the asymptotes is 2tan–1(b/a). If we draw lines through B, B parallel to the transverse axis and through A, A parallel to the conjugate axis, then P (a, b), Q (–a, b), R(–a, –b) and S(a, –b) all lie on the asymptotes x2/a2–y2/b2 = 0 so asymptotes are diagonals of the rectangle PQRS. This rectangle is called associated rectangle.
Note: x 2 y2 x 2 y2 x 2 y 2 H 2 2 1 , C 2 2 1 & A 2 2 = 0 b b b a a a clearly C + H = 2A {
H = hyperbola C = Conjugate hyperbola A = Asymptotes.}
RECTANGULAR OR EQUILATERAL HYPERBOLA A hyperbola is called rectangular if its asymptotes are at right angles. The asymptotes of x2/a2 –y2/b2 =1 are y = (b/a) x so they are perpendicular if –b2/a2 = – 1 i.e., b2 = a2, i.e., a = b. Hence equation of a rectangular hyperbola can be written as x2–y2 = a2 We give below some important observations of rectangular hyperbola. (a)
a2 = a2 (e2 – 1) gives e2 = 2 i.e., e = 2 .
(b)
Asymptotes are y = x.
(c)
Rotating the axes by an angle – / 4 about the same origin, equation of the rectangular hyperbola x2 – y2 = a2 is reduced to xy = a2/2 or xy = c2, (c2 = a2/2). In xy = c2, asymptotes are coordinate axes.
(d)
Rectangular hyperbola is also called equilateral hyperbola. Rectangular Hyperbola referred to its asymptotes as axis of co-ordinates
146
HYPERBOLA c , t R ~ (0}. t
(a)
Equation is xy = c2 with parametric representation x = ct, y =
(b)
Equation of a chord joining the points P (t1) & Q (t2), x + t1 t2y = c (t1 + t2)
(d)
Vertex of this hyperbola is (c, c) & (–c, –c) and focus is ( 2c,
2c ), ( 2c, 2c).
POSITION OF A POINT P w.r.t HYPERBOLA Let S = 0 be the hyperbola and P (x1, y1) be the point and S1 S(x1, y1). Then S1 < 0 P is in the exterior region S1 > 0 P is in the interior region
y Interior region
Exterior region
Interior region
O
x
S1 = 0 P lies on the hyperbola
LINE AND A HYPERBOLA The straight line y = mx + c is a secant, a tangent or passes outside the hyperbola
x 2 y2 1 according as : c2 > = < a2 m2 – b2. a 2 b2
TANGENT AND NORMAL Tangent (a)
xx yy x 2 y2 Equation of the tangent to the hyperbola 2 2 1 at the point (x1 y1) is 21 21 1 a b a b
(b)
x 2 y2 Equation of the tangent to the hyperbola 2 2 1 at the point ( a sec , b tan ) is a b x sec y tan 1 a b
Note : Equation of a chord joining 1 & 2 is
147
x y 2 2 cos 1 2 sin 1 cos 1 a 2 b 2 2
IIT-MATHEMATICS-SET-IV (c)
y = mx a 2 m 2 b 2 can be taken as the tangent to the hyperbola
x 2 y2 1 a 2 b2
Normal (a)
x 2 y2 The equation of the normal to the hyperbola 2 2 1 at point P(x1, y1) on the curve a b a 2x b2y = a2+ b2 a2e2 x1 y1
(b)
The equation of the normal at the point P (a sec , btan ) hyperbola
(c)
x 2 y2 ax by 2 1 is = a2 + b2 = a2 e2. 2 a b sec tan
In general, four normals can be drawn to a hyperbola from any point and if , , , be the concentric angles of these four co-normal points, then is an odd multiple of .
Chord of Contact of Tangents Drawn from a Point Outside the Hyperbola Chord of contact of tangents drawn from a point outside the hyperbola is T = 0 i.e., (xx1/a2) – (yy1/b2) = 1. From any point on the hyperbola
x 2 y2 1 tangents are drawn to the hyperbola a 2 b2
x 2 y2 2 . Then show that the area cut-off by the chord of contact on the asymptotes is 4 a 2 b2 ab
Solution : Let P (x1, y1) be a point on the hyperbola
x 2 y2 x12 y12 1 1 . . Then, a 2 b2 a 2 b2
The chord of contact of tangent from P to the hyperbola
x 2 y2 2 is a 2 b2
xx 1 yy1 2 2 ...(i) a2 b
The equations of the asymptotes are
x y x y 0 and 0 a b a b
The points of intersection of (i) with the two asymptotes are given by x1 =
2a 2b 2a 2b ,y , x2 ,y x1 y1 1 x1 y1 x 1 y1 2 x 1 y1 a b a b a b a b 148
HYPERBOLA
1 1 8ab | x y x y | Area of the triangle = 1 2 2 1 2 2 x 12 y12 2 2 b a
4ab
CHORD OF HYPERBOLA WITH SPECIFIED MID-POINT Chord of hyperbola with specified mid-point (x1, y1) is T = S1 , where S1 and T have usual meanings.
DIRECTOR CIRCLE The locus of the point of intersection of two perpendicular tangents to a hyperbola is called its director circle. Its equation is x2 + y2 = a2 – b2. Equation of any tangent to x2/a2– y2/ b2 = 1 is y = mx Tangent perpendicular to (i) is y = –
1 x m
a
2
/ m2 b 2
(a 2 m 2 b 2 )
... (i)
... (ii)
Locus of point of intersection of these perpendicular tangents i.e., equation of the director circle can be obtained by eliminating m between (i) and (ii).
(y – mx)2 + (my + x)2 = a2m2 – b2 + a2 – b2 m2 = (a2 – b2) (m2 + 1)
or (m2 + 1) x2 + (m2 + 1)y2
Cancelling (m2 + 1), we get the equation of director circle as x2 + y2 = a2 – b2.
HIGHLIGHTS Equation of the tangent at P (t) is
2.
Equation of the normal at P(t) is xt3 – yt = c (t4 – 1). where p is the point on the curve xy = c2.
3.
149
x + yt = 2c where p is the point on the curve xy = c2 t
1.
x2 y2 Locus of the feet of the perpendicular drawn from focus of the hyperbola 2 = 1 upon a b2 any tangent is its auxiliary circle i.e., x2 + y2 = a2 and the product of the feet of these perpendiculars is b2 .
4.
The portion of the tangent between the point of contact and the directrix subtends a right angle at the corresponding focus.
5.
The tangent and normal at any point of a hyperbola bisect the angle between the focal radii. This spells is reflection property of the hyperbola as an incoming light ray aimed towards one focus is reflected from the outer surface of the hyperbola towards the other focus.
6.
If from any point on the asymptote a straight line be drawn perpendicular to the transverse axis, the product of the segments of this line intercepted between the point and the curve is always equal to the square on the semi conjugate axis.
IIT-MATHEMATICS-SET-IV 7.
If the angle between the asymptote of a hyperbola
x 2 y2 1 is 2 , then the ecentricity of a 2 b2
the hyperbola is sec . 8.
If a circle intersects a rectangular hyperbola at four points, then the mean value of the points of intersection is the midpoint of the line joining the centres of circle and hyperbola.
9.
A rectangular hyperbola circumscribing a triangle passes through the orthocentre of this tri-
c angle. If ct i , t = i = 1, 2, 3 be the angular points P, Q, R then orthocentre is i c , ct1t 2 t 3 . ti t 2 t 3 10.
If a circle and the rectangular hyperbola xy = c2 meet in the four points t1 , t2, t3 & t4 , then (a) t1 t2 t3 t4 = 1 (b) the centre of the mean position of the four points bisects the distance between the centre of the two curves. (c) the centre fo the cirlce through the points t1, t2 & t3 is c 1 c1 1 1 t1 t 2 t 3 , t 1t 2 t 3 . t1 t 2 t 3 2 t 1 t 2 t 3 2
150
HYPERBOLA
151
IIT-MATHEMATICS-SET-IV
6-A
HYPERBOLA
152
HYPERBOLA ILLUSTRATION : 01
x2 y 2 x2 y2 1 2 1 and the hyperbola If the foci of the ellipse coincide, then the value of 25 b 144 81 25 b 2 is A) 3
B) 16
C) 9
D) 12
Solution : If eccentricities of ellipse and hyperbola are e & e1
foci ae ,0 & a1e1 ,0 Here ae a1e1 a 2e 2 a12e12 b12 b2 2 2 a 1 2 a1 1 2 a a1 2 2 2 a b a1 b12 144 81 25 b 2 =9 25 25 b 2 16
ILLUSTRATION : 02
x2 y 2 If PQ is a double ordinate of the hyperbola 2 2 1 such that OPQ is an equilateral triangle, O a b being the centre of the hyperbola. Then the eccentricity e of the hyperbola satisfies A) 1 < e <
2 3
2 3
B) e
C) e
3 2
Solution :
a P b 2 2 , b
M
O
a Q b2 2 , b
PQ is the doubled ordinate. Let MP MQ l given that
OPQ s an equilateral then OP = OQ = PQ
153
(OP) 2 (OQ )2 ( PQ) 2
D) e
2 3
IIT-MATHEMATICS-SET-IV
a2 2 2 a2 2 2 2 b l l b l l 2 4l 2 b2 b2
a2 2 2 b l 3l 2 2 b
a2 a2 l 2 3 2 b
l2
3b 2 a 2
e2
a 2b 2 0 3b2 a2
4 3
3b 2 a 2 0
3a 2 e 2 1 a 2
e
2 3
ILLUSTRATION : 03 The equations of the asymptotes of the hyperbola 2 x 2 5 xy 2 y 2 11x 7 y 4 0 are A) 2 x 2 5 xy 2 y 2 11x 7 y 5 0
B) 2 x 2 4 xy 2 y 2 7 x 11 y 5 0
C) 2 x 2 5 xy 2 y 2 11x 7 y 5 0
D) None of these
Solution : The pair of asymptotes curve differ by a constant.
Pair of asymptotes 2 x 2 5 xy 2 y 2 11x 7 y 0
……..……(1)
Hence (1) represents a pair of straight lines.
0 7 11 5 then 2 x 2 x l + 2 x - x x -2x 2 2 2
2
7 - 2 x 2
2
2
11 5 - 0 2 2
5 From (1) pair of asymptotes is 2 x 2 5 xy 2 y 2 11x 7 y 5 0 ILLUSTRATION : 04
x2 y2 If a variable straight line x cos y sin p , which is a chord of the hyperbola 2 2 1 b a , a b subtend a right angle at the centre of the hyperbola then it always touches a fixed circle whose radius is
A)
ab b 2a
B)
a a b
ab
ab
C)
b2 a2
D) b
b a 154
HYPERBOLA Solution : Since x cos y sin p subtends a right angle at centre i.e. (0,0). Making homogeneous equation of hyperbola
x2 y2 1 with the help of x cos y sin p and putting coefficient of x 2 coefa2 b2
ficient of y 2 0 . We get
1 1 1 2 2 2 a b p p
ab 2
b a2
p is also the length of perpendicular drawn from (0, 0) to the line x cos y sin p . ab
then radius of the circle = p =
b2 a2
P x cos y sin p
O M
ILLUSTRATION : 05 The equation of the line passing through the centre of a rectangular hyperbola is If one of its asymptote is 3x - 4y -6=0. The equation of the other asymptote is A) 4 x 3 y 8 0
B) 4 x 3 y 17 0
C) 3 x 2 y 15 0
D) None of these
x - y -1=0.
Solution : Since asymptotes of rectangular hyperbola are perpendicular to each other. Given asymptote is 3x 4 y 6 0
Other asymptotes is 4 x 3 y 0
……………….(1)
Given centre of hyperbola lies on x - y -1=0 since asymptotes pass through the centre of hyperbola
Centre is the point of intersection of x y 1 0 and 3x 4 y 6 0
155
IIT-MATHEMATICS-SET-IV
Centre is 2, 3 also 2, 3 lies on (1) then - 8 - 9 + = 0 = 17 Hence other asymptote is 4 x 3 y 17 0
[ from (1)]
ILLUSTRATION : 06
x2 y2 If e is the eccentricity of the hyperbola 2 2 1 and is angle between the asymptotes, then a b cos
A)
2
1 e e
1 B) 1 e
C)
1 e
D)None of these
Solution : b 2 a 2 e2 1
……………..(1)
since angle between asymptotes =
b 2 tan 1 a
tan
b 2 a
or
b 2 tan 1 a
cos
a 2 a 2 b2
1
=
b 1 a
1
2
y
=
y
e
2
1 e
b ax
x
y
bx a
ILLUSTRATION : 07
a sec , b tan and a sec , b tan are the ends of a focal chord of
x2 y 2 2 1 , then tan tan 2 2 2 a b
equals to 156
HYPERBOLA A)
e 1 e 1
B)
1 e 1 e
C)
1 e 1 e
D)
e 1 e 1
Solution : Equation PQ is
x y cos sin cos a 2 b 2 2
Its passes through (ae, 0)
e cos 0 cos 2 2
cos 2 1 e cos 2
cos cos 2 2 1 e 1 e cos cos 2 2
tan
(By componentdo & dividendo method)
1 e tan 2 2 1 e y
Pa sec , b tan
ae,0
x
Qa sec , b tan
ILLUSTRATION : 08
x2 y2 1 which pass through the point If m1 and m2 are the slopes of the tangents to the hyperbola 25 16 (6, 2) then A) m1 m2 Solution :
157
24 11
B) m1m2
20 11
C) m1 m2
48 11
D) m1m2
11 20
IIT-MATHEMATICS-SET-IV
x2 y2 1 25 16
Equation of tangent in terms of slope y mx or
y mx
2
25m
2
16
25m 2 16
it is passing through (6, 2) then
2 6m
2
25m 2 16
4 36m 2 24m 25m 2 16
11m 2 24m 20 0
m1 m2
24 20 , m1m2 11 11
ILLUSTRATION : 09 The vertices of the hyperbola 9 x 2 16 y 2 36 x 96 y 252 0 are A) 6,3 and 6,3
B) 6,3 and 2,3
C) 6,3 and 6, 3
D) None of these
Solution : Given Hyperbola is 9 x 2 16 y 2 36 x 96 y 252 0
9 x 2 4 x 16 y 2 6 y 252 0
9 x 2 36 16 y 3 144 252 0
9 x 2 16 y 3 144
or or
2
2
2
x 2 42
2
2
y 3 32
2
1
Vertices x 2 4 & y 3 0
i.e. 2 4, 3 or (6,3) &( 2,3
158
HYPERBOLA
y
4
2,3
2,3
6,3
x
2 ILLUSTRATION : 10 If a triangle is inscribed in a rectangular hyperbola, its orthocentre lies A) inside the curve C) on the curve
B) outside the curve D) None of these
Solution : Let rectangular hyperbola xy c 2
Let three points on (1) are
……………….(1)
c c c A ct1 , t , B ct2 , t , C ct3 , t 1 2 3
Let orthocentre is P(h, k) then slope of AP x slopes of BC = -1
c c c t3 t 2 t1 1 h ct1 ct3 ct2
k
k
c ht2t3 ct1t2 t3 t1
c similarly, BP AC then k t ht3t1 ct1t2 t3 2 c Subtracting (3) from (2) then we get h t t t 1 2 3
Substituting the value h in (2) then K = ct1t2 t3
c Orthocentre is t t t , ct1t2t3 123 which lies on xy c 2 159
c t1 1 1 h ct1 t 2 t3 k
…………….(2)
…………….(3)
IIT-MATHEMATICS-SET-IV
y
c ct1 , A t1 Ph , k
c ct 2 , t2 B
C c ct 3 , t3
***
160
HYPERBOLA
SECTION A - SINGLE ANSWER TYPE QUESTIONS 1.
If the foci of the hyperbola 9x2 - 16y2 + 18x + 32y - 151 = 0 are A) (2, 3), (5, 7)
2.
3.
4.
B) 4/ 3
B) e = 2/ 3
B) 1
C) 2/ 3
D) none of these
C) e =
3 /2
D) e > 2/ 3
C) 2
D) 3
B) parabolas
C) hyperbolas
D) None of these
C the centre of the hyperbola x2/a2 - y2 /b2 = 1. The tangent at any point P on this hyperbola meets the straight lines bx - ay = 0 and bx + ay = 0 in the points Q and R respectively. Show that CQ. CR = B) a2 - b2
C)
1 1 2 2 a b
D)
1 1 a 2 b2
x2 y2 Locus of point of intersection of two tangents to the hyperbola 2 2 1 , the product of whose a b 2 slopes is c the curve B) y2 + a2 = c2 (x2 - b2) D) y2 + b2 = c2 (x2 - a2)
The line y = x + 2 touches the hyperbola 5x2 - 9y2 = 45 at the point A) (0, 2)
161
1
If e and e’ be the eccentricities of two conics S and S’ such that e2 + e’2 = 3, then both S and S’ are
A) y2 - a2 = c2 (x2 + b2) C) y2 - b2 = c2 (x2 + a2) 10.
sin 2
If e() be the eccentricity of rectangular hyperbola xy = then the value of e() - e(6) is
A) a2 + b2
9.
y2
x2 y 2 If PQ is a double ordinate of the hyperbola 2 2 1 such that OPQ is an equilateral triangle, O a b being the centre of the hyperbola, then the eccentricity e of the hyperbola satisfies
A) ellipses 8.
x2
The eccentricity of the hyperbola whose latus-rectum is 8 and conjugate axis is equal to half the distance between the foci is
A) 0 7.
D) none
cos 2 A) eccentricity B) abscissa of foci C) directrix D) vertex 1 1 If e and e’ be the eccentricities of a hyperbola and its conjugate, then e 2 '2 e A) 0 B) 1 C) 2 D) None
A) 1 < e < 2/ 3 6.
C) (0, 0), (5, 3)
Which one of the following is independent of a in the hyperbola (0 < < /2)
A) 4/3
5.
B) (4, 1) (-6, 1)
B) (3, 1)
C) (-9/2, -5/2)
D) None
IIT-MATHEMATICS-SET-IV 11.
2x + 6 y = 2 touches the hyperbola x2 - 2y2 = 4, then the point of contact is 1 1 A) , 2 6
12.
C) (4,
6)
D) (-2,
6)
A common tangent to 9x2 - 16y2 = 144 and x2 + y2 = 9 is A) y =
13.
B) (4, - 6 )
3 15 x 7 7
B) y = 3
2 15 x 7 7
C) y = 2
3 x 15 7 D) none of these 7
If (a sec q, b tan q) and (a sec f, b tan f) are the ends of a focal chord of
x2 y 2 1 , then tan/2 tan a 2 b2
/2 equals A)
14.
e 1 e 1
B)
1 e 1 e
D)
e 1 e 1
x2 y 2 y2 x2 1 1 are and b2 b2 a2 b2
A) y = ± x ±
B) y = ± x ±
a 2 b2
D) y = ± x ±
a 2 b2
b2 a 2
The value of m for which y = mx + 6 is a tangent to the hyperbola
A) 16.
1 e 1 e
The equations to the common tangents to the two hyperbolas
C) y = ± x ± (a2 - b2)
15.
C)
17 20
B)
20 17
C)
x2 y 2 = 1 is 100 49
3 20
D)
20 3
If the tangent and normal to a rectangular hyperbola cut off intercepts a1 and a2 on one axis and b1 and b 2 on the other axis, then A) a1b1 a2b2 0
17.
C) a1a2 b1b2 0
D) none of these
x2 y2 If 2 2 1 (a > b) and x2 - y2 = c2 cut at right angles then a b A) a2 + b2 = 2c2
18.
B) a1b2 b2 a1 0
B) b2 - a2 = 2c2
C) a 2 b 2 2c 2
D) a 2b 2 2c 2
Let P (a sec , b tan ) and Q (a sec , b tan ), where + = /2, be two points on the hyperbola
x2 y 2 = 1. If (h, k) is the point of intersection of the normals at P and Q, then k is equal to a2 b2
162
HYPERBOLA
a2 b2 A) a 19.
23.
C) 0
B) 3y - 4x + 4 = 0
163
D) 3x - 4y = 2
C) a 2 / x 2 b 2 / y 2 (a 2 b 2 )2
D) None
A rectangular hyperbola whose centre is C is cut by any circle of radius r in four points P, Q, R and S. Then CP2 + CQ2 + CR2 + CS2 = B) 2r2
C) 3r2
D) 4r2
The diameter of 16x2 - 9y2 = 144 which is conjugate to x = 2y is 16 x 9
B) y =
32 x 9
C) x =
16 y 9
D) x =
32 y 9
If x = 9 is the chord of contact of the hyperbola x2 - y2 = 9, then the equation of the corresponding pair of tangents is B) 9x2 - 8y2 - 18x + 9 = 0 D) 9x2 - 8y2 + 18x + 9 = 0
If the sum of the slopes of the normals from a point P on hyperbola xy = c2 is constant k (k > 0), then the locus of P is B) x2 = kc2
C) y2 = ck2
D) x2 = ck2
If (a - 2) x2 + ay2 = 4 represents a rectangular hyperbola then a equals A) 0
28.
C) 4x - 4y = 3
B) x 2 / a 2 y 2 / b 2 (a 2 b 2 ) 2
A) y2 = k2c 27.
D) none
A) a 6 / x 2 b6 / y 2 (a 2 b2 ) 2
A) 9x2 - 8y2 + 18x - 9 = 0 C) 9x2 - 8y2 - 18x - 9 = 0 26.
D) none
The locus of the poles of normal chords of the hyperbola x2 /a2 - y2/a2 = 1 is
A) y = 25.
1 C) t2 t 2 1
The locus of the middle points of chords of hyperbola 3x2-2y2+4x-6y=0 parallel to y=2x is
A) r2 24.
1 B) t2 t 1
B) -1
A) 3x - 4y = 4 22.
a 2 b2 D) b
PQ and RS are two perpendicular chords of the rectangular hyperbola xy = c2. If C is the centre of this hyperbola then product of the slopes of CP, CQ, CR and CS is A) 1
21.
a 2 b2 C) b
If the normal at (ct1 , c / t1 ) on the curve xy = c2 meets the curve again at the point (ct2, c/t2) then 1 A) t2 t 3 1
20.
a 2 b2 B) a
B) 2
C) 1
D) 3
x2 y2 1 from where two perpendicular tangents can The number of point(s) outside the hyperbola 25 36
IIT-MATHEMATICS-SET-IV be drawn to the hyperbola is/are A) 1 29.
B) 2
C) infinite
The equations of the asymptotes of the hyperbola 2x2 + 5xy + 2y2 - 11x - 7y - 4 = 0 are A) 2x2 + 5xy + 2y2 - 11x - 7y - 5 = 0 C) 2x2 + 5xy + 2y2 - 11x - 7y + 5 = 0
30.
B) 2x2 + 4xy + 2y2 - 7x - 11y + 5 = 0 D) None of these
An ellipse has eccentricity 1/2 and one focus at the point P (1/2, 1). Its one directrix is the common tangent nearer to the point P, to the circle x2 + y2 = 1 and the hyperbola x2-y2 = 1. The equation of the ellipse in standard form is A) 9x2 + 12y2 = 108 C) 9 (x - 1/3)2 + 4( y - 1)2 = 36
31.
B) 9 (x - 1/3)2 + 12 (y - 1)2 = 1 D) None of these
The equation of the line passing through the centre of a rectangular hyperbola is x-y-1= 0. If one of its asymptote is 3x - 4y - 6 = 0, the equation of the other asymptote is A) 4x - 3y + 8 = 0
32.
D) zero
B) 4x + 3y + 17 = 0
If e is the eccentricity of the hyperbola
C) 3x - 2y + 15 = 0
D) none of these
x2 y 2 1 and is angle between the asymptotes, then cos/ a 2 b2
2= A) 33.
1 e e
1 e
D) None of these
C) 13 / 3
D) 2.5
C)
( x 1)2 ( y 5)2 1 D) None of these 16 9
The locus of the middle points of chords of hyperbola 3x2-2y2+4x-6y=0 parallel to y=2x is B) 3y - 4x + 4 = 0
C) 4x - 4y = 3
D) 3x - 4y = 2
The locus of the point of intersection of the line 3 x-y-4 3 k=0 and 3 kx + ky - 3 =0 is a hyperbola of eccentricity A) 1
37.
B) 1/2
( x 1)2 ( y 5)2 x2 y2 1 B) 1 16 9 16 9
A) 3x - 4y = 4 36.
C)
The equation of the hyperbola whose foci are (6, 5), (-4, 5) and eccentricity 5/4 is A)
35.
1 1 e
The eccentricity of the conic 4 (2y - x - 3)2 - 9 (2x + y - 1)2 = 80 is A) 2
34.
B)
B) 2
C) 2.5
D)
3
If a triangle is inscribed in a rectangular hyperbola, its orthocentre lies A) inside the curve
B) outside the curve
C) on the curve
D) none of these
164
HYPERBOLA 38. Equation of the hyperbola passing through the point (1, -1) and having asymptotes 0 and 3x + 4y + 5 = 0 is A) 3x2 - 10xy + 8y2 - 14x + 22y + 7 = 0 C) 3x2 - 10xy - 8y2 + 14x + 22y + 7 = 0 39.
B) (2, 2)
C) (-2, 4)
D) (1, 2)
C) x = h, y = h
D) x = k, y = k
The asymptotes of the hyperbola xy = hx + ky are A) x = k, y = h
42.
B) 9x2 - 8y2 - 18x + 9 = 0 D) 9x2 - 8y2 + 18x + 9 = 0
The points of intersection of the curves whose parametric equations are x=t2+1, y=2t and x = 2s, y = 2/ s is given by A) (1, -3)
41.
B) 3x2 + 10xy + 8y2 - 14x + 22y + 7 = 0 D) 3x2 + 10xy + 8y2 + 14x + 22y + 7 = 0
If x=9 is the chord of contact of the hyperbola x2 - y2 = 9, then the equation of the corresponding pair of tangents is A) 9x2 - 8y2 + 18x - 9 = 0 C) 9x2 - 8y2 - 18x - 9 = 0
40.
x + 2y + 3 =
B) x = h, y = k
If P ( x1 , y1 ), Q ( x2 , y2 ), R( x3 , y3 ) and S ( x4 , y4 ) are 4 concyclic points on the rectangular hyperbola xy = c2, the co-ordinates of the orthocentre of the DPQR are A) ( x4 y4 )
43.
B) ( x4 , y4 )
D) ( x4 , y4 )
The equation of a hyperbola, conjugate to the hyperbola x2 + 3xy + 2y2 + 2x + 3y = 0 is A) x2 + 3xy + 2y2 + 2x + 3y + 1 = 0 C) x2 + 3xy + 2y2 + 2x + 3y + 3 = 0
44.
C) ( x4 y4 )
B) x2 + 3xy + 2y2 + 2x + 3y + 2 = 0 D) x2 + 3xy + 2y2 + 2x + 3y + 4 = 0
If the tangent and normal to a rectangular hyperbola cut off intercepts x1 and x2 on one axis and y1 and y2 on the other axis, then A) x1 y1 x2 y2 0
45.
B) x1 y2 x2 y1 0
165
B) Ellipse D) Rectangular hyperbola
If locus of the centre of the circle which passes through (1, 2) and touches the circle +2x + 4y + 1 = 0 is A) Circle
47.
D) None of these
A conic has focus at (1, 2) and directrix is 3x+4y+4=0, and latus rectum is 6 then the conic is A) Parabola C) Non rectangular hyperbola
46.
C) x1 x2 y1 y2 0
B) Parabola
C) Ellipse
D) Hyperbola
The locus of the centre of the circle which passes through (2, -1) and touches the circle - 4x + 2y - 1 = 0 is
IIT-MATHEMATICS-SET-IV A) Circle 48.
The equation
B) Parabola
5 1 2
3
11 1 2
C)
13 1 2 3
D)
13 1 2 3
B) 3
C)
2
D) 2
B) 5
C) 7
D) 9
B) xy = 2a2
C) 2xy = a2
D) none
x2 y2 Tangents drawn from (c, d) to the hyperbola 2 2 = 1 make angles and with the x-axis. If a b 2 2 tan. tan = 1 then c – d = A) a2 – b2
55.
B)
If the axes are rotated through an angle of 450 in the anticlockwise direction then the equation of rectangular hyperbola x2 – y2 = a2 changes to A) xy = a2
54.
B) Ellipse D) Rectangular hyperbola
x y2 x2 y2 1 2 1 and that of the hyperbola The foci of the ellipse coincide, then the value 16 b 144 81 25 of b2 is A) 1
53.
D) Hyperbola
et e t et e t The locus of the point 2 , 2 is a hyperbola of eccentricity A)
52.
C) Ellipse
2x+3y-5=0
If the latus rectum of a hyperbola forms an equilateral triangle with the vertex at the centre of the hyperbola, then its eccentricity e =
A)
51.
D) Hyperbola
x y a represents a
A) Parabola C) Non rectangular hyperbola 50.
C) Ellipse
The locus of the centre of a circle which passes through (3, 2) and touches the line is A) Circle
49.
B) Parabola
B) b2 – a2
C) a2 + b2
D) none
x2 y 2 If the tangent at (, ) to the hyperbola 2 2 = 1 cuts the auxilary circle at points whose ordinates a b are y1 and y2 then 1/y1 + 1/y2 = A) 1/
B) 2/
C) 1/
D) 2/
166
HYPERBOLA 56.
P() and Q() are two points on
x2 y2 = 1 such that - = 2a. PQ touches the conic a 2 b2
x 2 cos 2 y 2 x 2 y 2 cos2 x2 y2 2 1 A) = 1 B) 2 C) 2 2 cos 2 2 2 a b a b a b 57.
The locus of middle points of normal chords of the rectangular hyperbola x2 – y2 = a2 is A) (x2 + y2)2 = 4a2x2y2 C) (x2 – y2)3 + 4a2x2y2 = 0
58.
B) (x2 + y2)3 = 4a2x2y2 D) (x2 – y2)2 + 4a x2 y2 = 0
The polar of a point on the ellipse x2 + y2 = 1 w.r.to hyperbola x2 - y2 = 1 touches conic A) x2 - y2 = 1
59.
D) none
B) x2 + y2 = 1
C) x2 + y2 = a/3
D) x2 - y2 =
A line L touches the circle on the line joining the foci of the hyperbola as diameter. The locus of pole of
x2 y 2 L w.r.to the same hyperbola 2 2 = 1 is a b
60.
A)
1 x2 y 2 2 = 2 2 a b2 a b
B)
1 x2 y 2 4 = 2 4 a b2 a b
C)
1 x2 y 2 4= 2 4 a b2 a b
D)
1 x2 y 2 4 = 2 4 a b2 a b
If the tangents and the normal to the hyperbola x2 – y2 = a2 cut of intercepts a1 and a2 on one axis and b1 and b2 on the other then a1a2 + b1b2 = A) a2
61.
B) outside the curve
D) none of these
B) 2x2 + 5xy + 2y2 – 11x – 7y = 3 D) 2x2 + 5xy + 2y2 – 11x – 7y + 3 = 0
B) 14x2 + 38xy + 20y2 - x + 7y + 9 = 0 D) none
The points of intersection of the asymptotes with the directrices lie on A) Director circle
167
C) on the curve
The asymptotes of the hyperbola are 7x+5y = 3 and 2x + 4y + 1 = 0. If it passes through (-2, 1), then its equation is A) 14x2 + 38xy + 20y2 + x – 7y + 9 = 0 C) 14x2 + 38xy + 20y2 – 7x + y – 9 = 0
64.
D) 0
The equation to the asymptotes of the hyperbola 2x2 + 5xy+2y2 + 11x – 7y – 4 = 0 is A) 2x2 + 5xy + 2y2 – 11x – 7y = 5 C) 2x2 + 5xy + 2y2 – 11x – 7y + 5 = 0
63.
C) 1
If a triangle is inscribed in a rectangular hyperbola, its orthocentre lies A) inside the curve
62.
B) a
B) Auxilary circle
C) Circle on SS1 as diameter
D) none
IIT-MATHEMATICS-SET-IV 65.
If m is a variable the locus of the point of intersection of the line A) Parabola
66.
B) Ellipse
x y x y 1 m and is a/an 3 2 3 2 m
C) Hyperbola
D) None of these
If the chords of contact of tangents from two points x1 , y1 and x2 , y2 to the hyperbola
x2 y 2 1 a 2 b2
x1 x2 are at right angles, then y y is equal to 1 2
a A) 2 b
67.
1 t3
B) t '
1 t
C) t '
1 t2
2 D) t '
1 t2
x2 y 2 x2 y 2 1 2 . The tangents are drawn to the hyperbola a 2 b2 a2 b2 area cut off by the chord of contact on the asymptotes is equal to B)
C) 2ab
D) 4ab
If PN is the perpendicular from a point on a rectangular hyperbola to its asymptotes, the locus, the midpoint of PN is A) circle
70.
a4 D) 4 b
From any point on the hyperbola
A) ab/2 69.
b4 C) 4 a
c If the normal at ct , on the curve xy c 2 meets the curve again in ‘t’ then t
A) t '
68.
b2 B) 2 a
B) parabola
C) ellipse
Let P a sec , b tan and Q a sec , b tan , where
D) hyperbola
, is the point of intersection of the 2
normals of P and Q, then k is equal to
a 2 b2 A) a 71.
72.
a 2 b2 B) a
a 2 b2 C) b 2
a 2 b2 D) b
If x 9 is the chord of contact of the hyperbola x 2 y 2 9, then the equation of the corresponding pair of tangents is A) 9 x 2 8 y 2 18 x 9 0
B) 9 x 2 8 y 2 18 x 9 0
C) 9 x 2 8 y 2 18 x 9 0
D) 9 x 2 8 y 2 18 x 9 0
A common tangent to 9 x 2 16 y 2 144 and x 2 y 2 9 is
168
HYPERBOLA A) y 73.
3 15 x 7 7
B) y 3
2 15 x 7 7
C) y 2
3 x 15 7 7
D) 3
2 15 x 7 17
A rectangular hyperbola whose centre is C is cut by any circle of radius r in the four
points P, Q, R
and S. Then CP 2 CQ 2 CR 2 CS 2 = A) r 2 74.
75.
If a sec , b tan and a sec , b tan tan
tan equals to 2 2
A)
e 1 e 1
The foci of the ellipse A) 1
77
D) 4r 2
x2 y2 are the ends of a focal chord of 2 2 1, then a b
1 e 1 e
C)
1 e 1 e
D)
e 1 e 1
B) y x a 2 b 2
C) y x a 2 b 2
D) y x a 2 b 2
x2 y2 x2 y 2 1 2 1 and the hyperbola coincide, then the value of b 2 is 16 b 144 81 25 B) 5
C) 7
D) 9
The curve represented by x a cosh sinh , y b cosh sinh is A) a hyperbola
78
B)
C) 3r 2
x2 y2 y 2 x2 1 The equation to the common tangents to the two hyperbolas 2 and 2 2 1 are a b2 a b A) y x b 2 a 2
76
B) 2r 2
B) an ellipse
C) a parabola
D) a circle
If P x1 , y1 ; Q x2 , y2 ; R x3 , y3 and S x4 , y4 are four concyclic points on the rectangularh y perbola xy c 2 , the coordinates of orthocentre of the PQR are A) x4 , y4
79
169
D) x4 , y4
B) 4
C) 6
D) 5
If e and e1 , are the eccentricities of the hyperbolas xy c 2 and x 2 y 2 c 2 then e2 e12 is equal to A) 2
81
C) x4 , y4
x2 y2 The number of normals to the hyperbola 2 2 1 from an external point is a b A) 2
80
B) x4 , y4
B) 4
C) 6
D) 8
The equation of the asymptotes of the hyperbola 2 x 2 5 xy 2 y 2 11x 7 y 4 0 are
IIT-MATHEMATICS-SET-IV
82
A) 2 x 2 5 xy 2 y 2 11x 7 y 5 0
B) 2 x 2 4 xy 2 y 2 7 x 11 y 5 0
C) 2 x 2 5 xy 2 y 2 11x 7 y 5 0
D) None of these
The combined equation of the asymptotes of the hyperbola 2 x 2 5 xy 2 y 2 4 x 5 y 0 is A) 2 x 2 5 xy 2 y 2 4 x 5 y 2 0
B) 2 x 2 5 xy 2 y 2 4 x 5 y 2 0
C) 2 x 2 5 xy 2 y 2 0
D) None of these
***
170
HYPERBOLA
KEY 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
b
b
b
d
c
a
c
a
d
c
d
16.
c
31.
b
46.
d
61.
17. 18. 19.
c
d
a
32. 33. 34.
c
c
a
47. 48. 49.
c
b
a
62. 63. 64.
c
c
76.
77. 78. 79.
c
171
a
a
d
b
b
20. 21. 22. 23. 24. 25. 26.
a
a
a
d
b
b
b
35. 36. 37. 38. 39. 40. 41.
a
b
c
b
b
b
a
50. 51. 52. 53. 54. 55. 56.
c
c
c
d
c
d
a
65. 66. 67. 68. 69. 70. 71.
c
d
a
80. 81. 82.
b
a
d
d
d
b
b
12. 13. 14. 15.
b
b
b
a
27. 28. 29. 30.
c
d
d
b
42. 43. 44. 45.
d
b
c
a
57. 58. 59. 60.
c
b
d
d
72. 73. 74. 75.
d
d
a
b
IIT-MATHEMATICS-SET-IV
SECTION B - MORE THAN ONE ANSWER TYPE QUESTIONS 1.
The equation of tangents to the hyperbola x2 - 4y2 = 36 which are perpendicular to the line x - y + 4 = 0 A) y = - x + 3 3
2.
D) none
B) (2, -1)
C) (-2, 1)
D) (-2, 1)
If the normal at any point P to the rectangular hyperbola x2 - y2 = 4 meets the axis in G and g and C is the centre of the hyperbola then A) PG = PC
4.
C) y = -x ± 2
Tangents are drawn to hyperbola x2 - y2 = 3 which are parallel to the line 2x + y + 8 = 0. Then their points of contact are A) (2, 1)
3.
B) y = - x - 3 3
B) Pg = PC
C) PG = Pg
D) Gg = 2PC
If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points P ( x1 , y1 ) , Q ( x2 , y2 ) , R ( x3 , y3 ) , S ( x4 , y4 ) then
5.
A) x1 x2 x3 x4 = 0
B) y1 y2 y3 y4 = 0
C) x1 x2 x3 x4 c 4
D) y1 y2 y3 y4 c 4
If m1 and m2 are the slopes of the tangents to the hyperbola x2/25 - y2/16 = 1 which pass through the point (6, 2) then A) m1 m2 24 / 11
6.
C) m1 m2 48 / 11
D) m1m2 11/ 20
The coordinates of a focus of the hyperbola 9x2 - 16y2 + 18x + 32y - 151 = 0 is A) (-1, 1)
7.
B) m1m2 20 / 11
B) (4, 1)
C) (4, 3)
D) (-6, 1)
A straight line touches the rectangular hyperbola 9x2 - 9y2 = 8 and the parabola y2=32x. The equation of the line is A) 9x + 3y - 8 = 0
B) 9x - 3y + 8 = 0
C) 9x + 3y + 8 = 0
D) 9x - 3y - 8 = 0
172
HYPERBOLA
KEY
173
1
2
ab
bc
3
4
abcd abcd
5
6
7
bc
bd
bc
IIT-MATHEMATICS-SET-IV
7
PROBABILITY
174
PROBABILITY
TERMS AND DEFINITIONS USED IN PROBABILITY : An experiment is called random if (i)
all the outcomes of the experiment are known in advance
(ii)
the exact outcome of any specific performance of the experiment is unpredicatable i.e. not known in advance. For Example drawing a card from a well shuffled pack of 52 playing cards is a random experiment.
Sample Space A set whose elements represent all possible outcomes of a random experiment is called the sample space and is usually represented by ‘S’. Consider the experiment of tossing a die. If we are interested in the number on the top face, then sample space would be S1 = {1, 2, 3, 4, 5, 6}. If we are interested only in whether the number is even or odd, then sample space is S2 = {even, odd}. Clearly more than one sample space can be used to describe the outcomes of an experiment. In this case ‘S1’ provides more information than ‘S2’. If we know which element in S1 occurs, we can tell which outcome in S2 occurs; however, knowledge of what happens in S2 in no way helps us to know which element in S1 occurs. In general it is desirable to use a sample space that gives the maximum information concerning the outcomes of the experiment.
Sample Point Each element of the sample space is called a sample point.
Event An event is a subset of the sample space. When a die is rolled, sample space is S = {1, 2, 3, 4, 5, 6}. Let A = {1, 3, 5}, B = {2, 4, 6}, C = {1, 2, 3, 4}. Here A is the event of occurrence of an odd number, B is the event of occurrence of an even number and C is the event of occurrence of a number less than 5.
Simple Event An event is called a simple event, if it is a singleton subset of the sample space S.
Compound Event A subset of the sample space S which contains more than one element is called a compound event
Equally likely Events A set of events is said to be equally likely if taking into consideration all the relevant factors there is no reason to expect one of them in preference to the others. For example when a fair coin is tossed, the occurrence of a tail or a head are equally likely.
Exhaustive Events A system of events is said to be exhaustive if on each performance of the experiment at least one of the events of the system is must to occur. In set theoretic notation, events A1, A2, . . . , m
Am are exhausitve if Ai S . For example on throwing of a die, the events {1, 2}, {2, 3, i 1
4}, {5} and {4, 5, 6} form an exhaustive system of events. 175
IIT-MATHEMATICS-SET-IV
Mutually Exclusive Events A set of events is said to be mutually exclusive if the occurrence of one of them precludes the occurrence of any of the remaining events. For example, when we throw a pair of dice, the events “ a sum of 5 occurs”, “a sum of 7 occurs” and “a sum of 9 occurs” are mutually exclusive. In set theoretic notation, events A1, A2, . . . , Am are mutually exclusive if Ai A j for i j and 1 i, j m .
CLASSICAL DEFINITION OF PROBABILITY If there are n exhausitve mutually exclusive and equally likely out comes of an experiment and m of them are favourable to an event A, then the probability of the happening of A is equal to
m and it is denoted by P(A). Clearly P(A) is a non-negative number not greater than unity.. n So 0 P(A) 1 . If probability of happening of an event A is 1, then A is called certain event and if probability of happening of event A is zero, then A is called impossible event.
Odds in Favour and Odds Against an Event As a result of an experiment, if p of the outcomes are favourable to an event E and q of the outcomes are against it, then we say that odds are p to q in favour of E or odds are q to p against E.
number of favourable cases
p
Odds in favour of an event E = number of unfavourable cases q
and odds against an event E =
number of unfavourable cases q number of favourable cases p
If odds in favour of an event E are p : q then the probability of the occurrence of that event is
p pq q Similarly the probability of the non-occurrence of that event is p q
SET THEORETIC PRINCIPLES If ‘A’ and ‘B’ be any two events of the sample space, then AÈ B would stand for occurrence of atleast one of them and AÇB stands for simultaneous occurrence of A and B. A or A stands for non–occurrence of A. (or (A¢ Ç B¢) stands for non–occurrence of both A and B. A Í B stands for ‘the AB occurrence of B implies the occurrence of A. If A and B are any two events, then P(AÈB) = P(A) + P(B) – P (AÇB). If A and B are mutually exclusive, P(AÈB) = P(A) + P(B). Hence P A = P(A¢) = 1 – P(A)
176
PROBABILITY P(AÇB¢) = P(A) – P(AÇB) Now P(exactly one of A and B occurs) = P A B P A B = P(A) + P(B) 2P(A B) = P(A B) P(A B)
Some Theorems 1. If A B , then (i) P A P B and (ii) P(B – A) = P(B) – P(A) 2. P A B C P A P B P C P A B P B C
P C A P A B C n P 3. Ai i1
n
P Ai and equality holds if and only if events A , i = 1, 2, . . . , n are i
i 1
exclusive. n P 4. Ai 1 and equality holds if and only if events Ai, i = 1, 2, . . . n are exhaustive. i 1 n
5.
P Ai = 1 if events A , i = 1, 2, . . ., n are exclusive and exhaustive. i 1
i
n n Note : If events A1, A2, . . . , An are exclusive, then P A i P Ai . This is called the i 1 i 1 rule of sum
INDEPENDENT EXPERIMENTS If two random experiments are performed and their outcomes are independent of each other, then the experiments are called independent experiments. For example (i) Consider the tossing of a coin twice, clearly the outcome for the second toss is not effected by the result of the first toss. So the two tosses are independent random experiments. (ii) Consider the drawing of two balls one after the other, with replacement, from an urn containing two or more balls. Then the two draws are independent of each other. So considering each draw as an experiment, the two experiments are independent random experiments. (iii) In throwing of a die and a coin together or one after the other are independent experiments.
Remarks (i) In drawing of two cards, without replacement, from a well-shuffled ordinary pack of 52playing cards, the two draws are not independent experiments. 177
IIT-MATHEMATICS-SET-IV (ii) Let E1 and E2 be two independent random experiments.Let A be an event of experiment E1 only and B be an event of experiment E2 only. Then P(A occurs in E1 and B occurs in E2) = P(A). P(B). For example if a coin and a die are thrown together and A = {H}, B = {1, 2, 3, 4}, then
1 4 1 P(head on coin and a number 4 on die) = P(A). P(B) = . 2 6 3
BINOMIAL DISTRIBUTION FOR SUCCESSIVE EVENTS Suppose p and q are the respective probabilities of the happening and failing of an event at a single trial. Then the probability of its happening r times (exactly) in n triails is nCr pr qn – r, because the probability of its happening r times and failing n – r times in a given order is prqn –r and there are nCr such orders, which are mutually exclusive and for any such order the probability is prqn – r. For example if a die is thrown five times and we want the probability of occurrence of a
2 1 1 2 , q 1 and n = 5, r = 4 6 3 3 3
composite number four times, then we have p = 4
n
r n– r
Thus required probability = Crp q
1
10 1 2 = C4 . 3 3 243 5
CONDITIONAL PROBABILITY The probability of occurrence of an event B when it is known that some event A has already occurred, is called the conditional probability and is denoted by P(B/A). The symbol P(B/A) is usually read as probability of B, given A. Consider two events A and B. When it is known that event ‘A’ has occurred, it means that sample space would reduce to that sample points representing event A. Now for P(B/A) we must look for the sample points representing the simultaneous occurrence of A and B i.e., sample points in A B . n A B n A B P(A B) n(S) P(B/A) = n(A) n A P(A) n(S)
Thus P(B/A) =
P A B , where 0 < P(A) 1 P A
Similarly, P(A/B) =
P A B P B
, 0 < P(B) 1
P(A).P(B / A), P(A) 0 Hence, P A B = P(B).P(A / B), P(B) 0
Independent Events
178
PROBABILITY Two events A and B are said to be independent if occurrence or non–occurrence of one does not affect the occurrence or non–occurrence of the other, i.e., P(B/A) = P(B), P(A) 0.
P A B P(B) P(A)
P(B/A) =
P(A B) = P(A). P(B) If the events are not independent, then they are said to be dependent.
Mutually independent Events Three events A, B and C are said to be mutually independent if,
P A B = P(A). P(B), P A C = P(A). P(C), P B C = P(B). P(C) and P A B C = P(A). P(B). P(C) These events would be said to be pair–wise independent if,
P A B = P(A). P(B), P B C = P(B). P(C) and P A C = P(A). P(C). Thus mutually independent events are pair–wise independent but the converse may not be true. P1, P2, . . . , P8 an eight players participating in a tournament. If i < j, then Pi will win the match against Pj. Players are paired up randomly for first round and winners of this round again paired up for the second round and so on. Find the probability that P4 reaches in the final.
Total Probability Theorem Suppose A1, A2 . . . , An are mutually exclusive and exhaustive events, then for any event B, n
we can write B =
B A i 1
n
P B P B Ai , as events B Ai , i = 1, 2, . . ., n are exclusive. i 1
n
P(B) =
P B / Ai .P Ai . i 1
Baye’s Theorem Suppose A1, A2, . . ., An are mutually exclusive and exhaustive set of events. Then the conditional probability that Ai happens (given that B has happened) is given by
179
IIT-MATHEMATICS-SET-IV P(Ai/B) =
P Ai B P A i .P B / A i n P B P Ai .P B / Ai i 1
GEOMETRICAL PROBABILITY If the number of points in the sample space is infinite, then we can not apply the classical definition of probability. For instance, if we are interested to find the probability that a point selected at random in a circle of radius r, is nearer to the centre then the circumference, we can not apply the classical definition of probability. In this case we define the probability as follows : P=
Measure of the favourable region , Measure of the sample space
where measure stands for length, area or volume depending upon whether S is one-dimensional, two – dimensional or three – dimensional region. Thus the probabiloity, that the chosen point is nearer to the centre than the circumference = 2
area of the region of the favourable point r / 2 1 = . area of the circle 4 r 2
180
PROBABILITY
181
IIT-MATHEMATICS-SET-IV
7-A
PROBABILITY
182
PROBABILITY
WORKEDOUT ILLUATRATION ILLUSTRATION : 01 If from each of the three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, one ball is drawn at random, then the probability that 2 white and 1 black ball will be drawn is A) 13/32
B) 1/4
C) 1/32
D) 3/16
Solution : The contents of three boxes are I
3W
1B
II
2W
2B
III
1W
3B
Let Wi i 1,2,3 be the event of drawing a white ball from i th box and B i i 1,2,3 be the event of drawing a black ball from 0th box. Then, Required probability.
P W1 W2 B 3 W1 B 2 W3 B1 W2 W3 P W1 W2 B 3 PW1 B 2 W3 P B1 W2 W3 P W1 P W2 P B 3 P W1 PB 2 P W3 P B1 P W2 P W3
3 2 3 3 2 1 1 2 1 26 13 . 4 4 4 4 4 4 4 4 4 64 32
ILLUSTRATION : 02 A fair coin is tossed repeatedly. If tail appears on first four tosses, then the probability of appearing on fifth toss equals A) 1/2
B) 1/32
C) 31/32
head
D) 1/5
Solution : Since the trials are independent so the probability that head appears on the fifth toss does not depend upon previous results of the tosses. Hence, required probability = probability of getting head = 1/2. ALITER Let A be the event of getting tail in first four tosses and B be the event of getting head on
fift h
4
1 1 1 toss. Then, PA , P B 1 1 1 1 and PA B = probability of getting tail in 2 2 2 5
1 first four tosses and head in fifth toss = . 2 Here, getting head on fifth toss means getting head on fifth toss and head or tail in first four tosses. Now, Required probability P B / A
183
P A B 1/ 25 1 P A 1/24 2
IIT-MATHEMATICS-SET-IV ILLUSTRATION : 03 There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Thenthe probability that only two tests are needed is A) 1/3
B) 1/6
C) 1/2
D) 1/4
Solution : The total number of ways in which two machines can be chosen out of four machines is 4 C 2 6. If only two tests are required to identify faulty machines, then in first two tests faulty machines are identified. This can be done in one way only. So, favourable number of ways =1 Hence, required probability = 1/6. ILLUSTRATION : 04 There are n persons sitting in a row. Two of them are selected at random. The pr o babilit y that two selected persons are not together is n n 1 A) 2/n B) 1-2/n C) D) None of these n 1n 2 Solution : n The total number of ways of selecting two persons out of n is C 2
n n 1 . 2
The number of ways in which two selected persons are together is n 1 . So, the probability that the
n 1 2 . n n 1 n selected persons are together is 2 Hence, required probability 1
2 . n
ILLUSTRATION : 05 If four dice are thrown together, then the probability that the sum of the numbers them is 13, is 5 11 35 1 A) B) C) D) 216 216 324 432
appearing on
Solution Total number of elementary events associated to the random experiment of throwing 4 dice
is
6 6 6 6 64
Favourable number of elementary events.
Coeff . of x 13 in x x 2 x 3 ........ x 6
Coeff . of x 9 in 1 x x 2 ...... x 5
4
4
184
PROBABILITY Coeff . of x
9
1 x 6 in 1 x
4
1 x in 1 4x ...... 1 x 4
Coeff . of x 9 in 1 x 6
Coeff . of x 9
4
6
4
Coeff. of x 9 in 1 x 4 4 coeff. of x 3 in 1 x 4
9 4 1 C 4 1 4 3 41 C 41
Coeff. of x
n
in 1 x r n r 1 C r 1
12 11 10 6 5 4 4 = 220-80=140. 3 21 3 2 1
12 C 3 4 6 C 3
Hence, required probability
140 35 64 324
ILLUSTRATION : 06 Six faces of a die are marked with the numbers 1,1,0,2,2 and 3. The die is thrownthrice. The probability that the sum of the numbers thrown is six, is 1 1 5 1 A) B) C) D) 72 12 108 36 Solution : Total number of elementary events 6 3 Favourable number of elementary events.
Coeff . of x 6 in x x 1 x 0 x 2 x 2 x 3
Coeff . of x
6
1 x x in
2
x3 x4 x5 x2
Coeff . of x 12
3
3
6
3
185
3
1
6 3 1
3
3
Coeff . of x 12 in 1 x 12 31 C 31 3 C1.
3
1 x in 1 C x C x
Coeff . of x 12 in 1 x 6
3
Coeff . of x 12 in 1 x x 2 x 3 x 4 x 5
1 x 6 Coeff . of x 12 in 1 x
2
3
12
3
......... 1 x
3
C1. Coeff .of x 6 in 1 x 3 C 2 . Coeff .of x 0 in 1 x
C 31 3 C 2
3
IIT-MATHEMATICS-SET-IV 14 C 2 3 C 1 8 C 2 3 C 2 91 84 3 10 Hence, required probability
10 5 3 108 6
ILLUSTRATION : 07 1 1 1 , , respectively. A box is s e 5 6 7 lected at random and a screw drawn from it at random is found to be defective. Then the probability that it came from box A is
The chances of defective screws in three boxes A, B, C are
A) 16/29
B) 1/15
C) 27/59
D) 42/107
Solution : Let E1,E 2 and E 3 denote the events of selecting box A,B,C respectively and A be the that a screw selected at random is defective.
event
Then PE1 PE 2 PE 3 1/ 3 PA / E 1
1 1 1 , PA / E 2 , PA / E 3 , 5 6 7
By Baye’s rule, Required probability PE 1 / A
PE1 PA / E1 PE1 P A / E1 P(E 2 )P A / E 2 PE 3 P A / E 3
1 1 42 3 5 1 1 1 1 1 1 107 3 5 3 6 3 7
ILLUSTRATION : 08 A bag contains 16 coins of which two are counterfeit with heads on both sides. The rest a r e fair coins. One is selected at random from the bag and tossed. The probability of getting a head is. A) 9/16
B) 11/16
C) 5/9
D) None of these
Solution : Let A be the event of selecting a counterfeit coin and B the event of getting head. Then, Required probability P A B P A B PA B PA B PA PB / A A PB / A
2 14 1 9 1 16 16 2 16 186
PROBABILITY ILLUSTRATION : 09 A fair coin is tossed 100 times. The probability of getting tails an odd number of times is A) 1/2
B) 1/8
C) 3/8
D) None of these
Solution : Let X denote the number of tails. Then, X is a binomial variate with parameters n=100and p=1/2. 1 100 Therefore, PX r Cr 2
100
, r 0,1,2,.......,100.
Required probability
P X 1 P X 3 ...... PX 99 100
1 2
100
C1 100 C 3 ..... 100 C 99
1 100
2
2 12 99
ILLUSTRATION : 10 A biased coin with probability p, 0
B) 2/3
C) 2/5
D) 3/5
Solution : Let q 1 p. Since head appears first time in an even throw i.e., 2 or 4 or 6 etc., 2/5 = qp q 3 p q 5 p .....
1 pp 1 p qp 2 2 5 1 q 2 5 1 1 p 2 2 p
22 p 51 p p 1/3
ILLUSTRATION : 11 A bag contains n+1 coins. It is known that one of these coins shows heads on both sides, whereas the other coins are fair. One coin is selected at random and tossed. If the pr o babilit y that toss results in heads is 7/12, then the value of n is. A) 3
B) 4
C) 5
D) None of these
Solution : Let E 1 , denote the event “ a coin with two heads is selected and E 2 , denote the event “ a coin is selected”. Let A be the even “the toss results in heads”. Then,
PE1 and
187
1 n , PE 2 , PA / E 1 1 n 1' n 1'
P A / E 2 1/ 2.
fair
IIT-MATHEMATICS-SET-IV
PA PE1 PA / E 1 PE 2 PA / E 2
7 1 n 1 1 12 n 1 n 1 2
12 6n 7n 7 n 5.
PA 7 / 12
ILLUSTRATION : 12 The probabilities that a student passes in Mathematics, Physics and Chemistry are m,p and c, respectively. Of these subjects, the student has 75% chance of passing in at least one, a 50% chance of passing in at least two, and a 40% chance of passing in exactly two. Which of the following relations are true. A) p m c
19 20
B) p m c
27 20
C) pmc
1 10
D) pmc
1 4
Solution : We have, PA B C
3 4
i.e. P A PB PC P A B P B C PA C PA B C PA B P B C P A C 2PA B C
1 2
……. (ii)
2 5
……. (iii)
3 4
and PA B P B C P A C 3P A B C
From (ii) and (iii), we get P A B C
1 2 1 2 5 10
PA PB PC
……. (iv)
1 1 pmc . 10 10
From (i) , (ii) and (iii), we have
188
PROBABILITY
SECTION A -SINGLE ANSWER TYPE QUESTIONS 1.
A card is drawn at random from a pack of cards. The prob. of this card being a red or queen is A) 1/3
2.
189
B) 5/11
C) 6/11
D) 1/6
B) 1/12
C) 1/9
D) None of these
B) 1
C) 23/24
D) 9/2
B) 1/18
C) 1/36
D) None of these
B) 4. (2/7)4
C) (3/7)3
D) None of these
B) (8/15)7
C) (3/5)7
D) None of these
Your are given a box with 20 cards in it. 10 of these cards have the letter I printed on them. The other ten have the letter T printed on them. If you pick up 3 cards at random and keep them in the same order, the probability of making the word IIT is A) 9/80
11.
D) None of these
15 coupons are numbered 1, 2, 3, ........., 14, 15 seven coupons are selected at random, one a time with replacement. The probability that 9 would be the largest number appearing on the selected coupon is A) (1/16)6
10.
C) 1/3
Seven chits are numbered 1 to 7. Three are drawn one by one with replacements. The probability that the least number on any selected chit is 5, is A) 1 - (2/7)4
9.
B) 3/5
Three identical dice are rolled. The probability that the same number will appear on each of them is A) 1/6
8.
D) 0
There are 4 addressed envelopes and 4 letters. Then the chance that all the letters are not mailed through proper envelope is A) 1/24
7.
C) 2/11
Two dice are thrown simultaneously. The probability of obtaining a total score of 5 is A) 1/18
6.
B) 4/11
In a throw of a dice the probability of getting one in even number of throws is A) 5/36
5.
D) 7/13
Three mangoes and three apples are in box. If two fruits are chosen at random, the probability that one is a mango and the other is an apple is A) 2/3
4.
C) 1/2
A single letter is selected at random from the word “PROBABILITY”. The probability that it is a vowel is : A) 3/11
3.
B) 1/26
B) 1/8
C) 4/27
D) 5/38
‘A’ draws two cards with replacements from a pack of 52 cards and ‘B’ throws a pair of dice what is the change that ‘A’ gets both cards of same suit and ‘B’ gets total of 6
IIT-MATHEMATICS-SET-IV A) 1/144 12.
B) 7/36
C) 1/4
D) None of these
B) 7/13
C) 5/13
D) 7/10
B) 1/15
C) 1/5
D) 14/15
B) 3 : 1
C) 2 : 3
D) None of these
B) 1/8
C) 1/6
D) None of these
B) 4/35
C) 4/33
D) 4/1155
Three are 3 works. One is of 3 volumes, one is of 4 volumes and one is of only one and they are placed a random in at shelf. What is the chance that volume of the same work is placed together A) 1/40
21.
D) p = 1, q = 1/2
If 3 distinct numbers are chosen randomly from {1, 2, ....., 100}, then probability that all are divisible by both 2 and 3 is A) 4/25
20.
C) p = 1, q = 0
A number is chosen at random among the first 120 natural numbers. The probability of the number chosen being a multiple of 5 or 15 is A) 1/5
19.
B) p = 1 = 1/2
One of the two events must occur. If the chance of one is 2/3 of the other, then odds in favour of the other are A) 1 : 3
18.
D) 65/81
Two numbers are chosen from {1, 2, 3, 4, 5, 6} one after another without replacement. Find the probability that one of the smaller value of two is less than 4. A) 4/5
17.
C) 80/81
Of cigarette smoking population 70% are men and 30% are women, 10% of these men and 20% of these women smoke Wills. Probability that a person seen smoking a Wills to be male is A) 1/5
16.
B) 1/81
Two dice are thrown simultaneously. The probability of obtaining a total score of seven is A) 1/6
15.
D) 7/144
A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II and III are p, q and 1/2 respectively. If the probability that the student is successful is 1/2, then A) p = q = 1
14.
C) 5/144
An unbaised die with faces marked 1, 2, 3, 4, 5 and 6 is rolled four times. Out of four face values obtained, the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is then A) 16/81
13.
B) 1/4
B) 3/140
C) 9/70
D) None of these
From a pack of 52 cards the cards are drawn till an ace appears. The probability that an ace does not come in first 26 cards is
190
PROBABILITY A) 46/153 22.
191
B) 1/6
C) 2/3
D) 5/9
B) 5/1944
C) 5/2592
D) None of these
B) 1 - (0.7)10
C) (0.7)10
10 times, the
D) (0.3)10
B) [1 - (1 - p)n]/n2
C) [1 - (1 - p)n]/n
D) none of these
B) p + q - 2pq
C) p + q - pq
D) p + q + pq
B) 24/150
C) 49/150
D) 56/150
Two numbers are chosen from {1, 2, 3, 4, 5, 6} one after another without replacement. Find the probability that one of the smaller value of two is less than 4 A) 4/5
31.
D) 1/20
The probability that Krishna will be alive 10 years hence is 7/15 and that Hari will be alive is 7/10. What is the probability that both Krishna and Hari will be dead 10 years hence A) 21/150
30.
C) 1/10
If the probabilities that A and B will die within a year are p and q respectively then the probability that only one of them will be alive at the end of the year is A) p + q
29.
B) 1/5
The probability that a man aged 50 years will die in a year is p. The probability that out of n men A1, A2,....,An each aged 50 years, A1 will die and be first to die is A) 1 - (1 - p)n
28.
D) 2/3
The probability that a person will hit a target in a shooting practice is 0.3. If he shoots probability that he hit the target is A) 1
27.
C) 1/2
An ordinary cube has four blank faces, one face marked 2, another marked 3. Then the probability of obtaining a total of exactly 12 in 5 throws is A) 5/1296
26.
B) 1/4
The probability that in the toss of two dice we obtain an even sum or a sum less than 5 is A) 1/2
25.
D) None of these
Three of the six vertices of a regular hexagon are chosen at random. The possibility that the triangle with three vertices is equilateral, equals A) 1/2
24.
C) 109/153
The probability of India winning a test match against West Indies is 1/2. Assuming independence from match to match, the probability that in a 5 match series India’s second win occurs at third test is A) 1/8
23.
B) 23/27
B) 1/15
C) 1/5
D) 14/15
The probability that a certain beginner at golf gets a good shot if he uses the correct club is 1/3, and the probability of a good shot with an incorrect club is 1/4. In his bag are 5 different clubs, only one of which is correct for the shot in question. If he chooses a club at random and takes a stroke, the probability that he gets a good shot is
IIT-MATHEMATICS-SET-IV A) 1/3 32.
D) 7/12
B) 3/4
C) 3/7
D) 37/56
A cricket has 15 members, of whom only 5 can bowl. If the names of the 15 members are put in6to a hat and 11 drawn at random, then the chance of obtaining an eleven containing at least 3 bowlers is A) 7/13
34.
C) 4/15
A purse contains 4 copper coins, 3 silver coins, the second purse contains 6 copper coins and 2 silver coins. A coin is taken out of any purse, the probability that it is a copper coin is A) 4/7
33.
B) 1/12
B) 11/15
C) 12/13
D) None of these
On a toss of two dice, A throws a total of 5. Then the probability that he will throw another 5 before he throws 7 is A) 1/9
B) 1/6
C) 2/5
D) 5/36
35.
A, B, C in order toss a coin. The first one to throw a head wins. Assuming the game continues indefinite their respective chances of winning the game are 4 2 1 1 4 2 2 4 1 A) , , B) , , C) , , D) None of these 7 7 7 7 7 7 7 7 7
36.
The mean number of heads in three tosses of a coin is A) 3/2
37.
C) < 1/2
D) 2/3
B) 1/2
C) 1/9
D) 1/10
B) 10/19
C) 5/19
D) None of these
A and B throw alternately with a pair of dice. A wins if he throws 6 before B throws he throws 7 before A throws 6. If A begins, find his change of winning. A) 30/16
41.
B) > 1/2
In a purse there are 9 five paisa coins and one rupee coin. In another purse there are all 10 five paisa coins. 9 coins are taken from the former and put into the second and then 9 coins are taken from the latter and put into the first. What is the change that rupee coin is still in the first purse A) 1/9
40.
D) 1/8
If the letters of the word REGULATION be arranged at random, the probability that there will be exactly four letters between R and E is A) 1/5
39.
C) 1/2
Two persons, A and B, have respectively n + 1 and n coins, which they toss simultaneously. Then the probability that A will have more heads than B is A) 1/2
38.
B) 5/2
B) 30/61
C) 13/60
7, and B if
D) 31/61
Ten pairs of shoes are in a closed. Four shoes are selected at random. Find the probability that there is at least one pair among the four selected. A) 99/323
B) 99/223
C) 124/323
D) None
192
PROBABILITY 42. The probability that a man will live 10 more years is 1/4 and the probability that his wife will live 10 more years is 1/3. Then the probability that neither will be alive in 10 years is A) 5/12 43.
B) 49/101
B) 1/3
B) > 1/4
B) 1 - P ( A / B )
B) .61
C) 50/101
C) 0
C) 1/2
C)
1 P( A B ) P( B )
C) .39
D) 51/101
D) 1
D) none of these
D)
P( A ) P( B )
D) None of these
B) P (Ac - Bc) = P (Ac) - P(Bc) D) P (A/B) = P(A)
If E and F are independent events such that 0 < P (E) < 1, and 0 < P (F) < 1. Then A) E and F are mutually exclusive B) E and Fc (complement of the event F) are independent C) Ec and Fc are independent
193
D) 0.0250
Let 0 < P (A) > 1, 0 < P (B) < 1 and P (A B) = P (A) + P(B) - P (A). P(B). then, A) P (B/A) =P(B) - P(A) C) P (A È B)c = P(Ac). P(Bc)
51.
C) 0.0625
Two events A and B have probability .25 and .50. The probability that both occur simultaneously is 14. Then probability that neither A nor B occur is A) .75
50.
B) .0875
If A and B are such events that P (A) > 0 and P (B) ¹ 1 then P (A / B ) is equal to A) 1 - P (A/B)
49.
D) 32/91
For any two independent events E1 and E2, P {E1 È (E2) Ç ( E 1 ) Ç ( E 2 ) } is] A) 1/4
48.
C) 51/91
A and B are two independent events. The probability that both A and B occur is 1/6 and the probability that neither of them occurs is 1/3. The probability of A is A) 1/2
47.
B) 44/91
One hundred identical coins, each with probability p, of showing up heads are tossed. If 0< p < 1 and the probability of heads showing on 50 coins is equal to that of the heads showing in 51 coins, then the value of p is A) 1/2
46.
D) 11/12
India plays two matches each with West Indies and Australia. In any match the probabilities of India getting points 0, 1 and 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability if India getting at least 7 points is A) 0.8750
45.
C) 7/12
Odds 8 to 5 against a person who is 40 years old living till he is 70 and 4 to 3 against another person now 50 till he will be living 80. Probability that one of them will be alive next 30 years A) 59/91
44.
B) 1/2
IIT-MATHEMATICS-SET-IV D) P (E/F) + P (Ec/F) = 1 52.
53.
If A and B are any two events, the probability that exactly one of them occurs is A ) P (A ) + P(B) - 2P (A B)
B) P ( A ) +P ( B ) -2P( A B )
C) P (A B ) + P( A B)
D) P (A) + P(B) - P (A B)
Let E and F be two independent events. The probability that both E and F happen is 1/12 and the probability that neither E nor F happens is 1/2. Then A) P(E) = 1/3, P(F) = 1/4 C) P(E) = 1/6, P(F) = 1/2
54.
B) P(E) = 1/2, P(F) = 1/6 D) P(E) = 1/4, P(F) = 1/3
For any two events A and B in a sample space A) P(A/B)
P( A ) P( B ) 1 ,P( B ) 0 is always true. P( B )
B) P (A B ) = P(A) - P(AB) does not hold C) P (AB) = 1 - P ( A ) P ( B ), if A and B are independent D) P (A B) = 1 - P ( A ) P( B ), if A and B are disjoint 55.
For n independent events Ai s, P(Ai)=1/(1+i), i=1,2...,n. The probability that atleast one of the events occurs is A) 1/n
56.
C) 0.80
D) 0.20
B) 9/25
C) 16/25
D) 24/25
B) 1/8
C) 3/4
D) 3/8
A number is chosen at random from among the first 30 natural numbers. The probability of the number chosen being a prime, is A) 1/3
60.
B) 0.85
A mans is known to speak truth in 75% cases. If he throws an unbiased die and tells his friend that it is a six, then the probability that it is actually a six, is A) 1/6
59.
D) none of these
The probability that a teacher will give an unanounced test during any class meeting is 1/5. If a student is absent twice, the probability that he will miss at least one test, is : A) 7/25
58.
C) n/(n + 1)
The probabilities that a student will obtain grades A,B,C or D are 0.30, 0.35, 0.20 and 0.15 respectively. The probability that he will receive atleatst C grade, is A) 0.65
57.
B) 1/(n + 1)
B) 3/10
C) 1/30
D) 11/30
Three players A, B, C in this order, cut a pack of cards, and the whole pack is reshuffled after each cut. If the winner is one who first draws a diamond then C’s chance of winning is 194
PROBABILITY A) 9/28 61.
1 nn
B) 1/12
C) 1/84
D) 19/84
B) Rs. 7: Rs. 4
C) Rs. 5.50; Rs. 5.50
D) Rs. 5.75; Rs. 5.25
B)
1 n!
C)
( n 1) ! n n 1
D) None of these
B) 2/19
C) 1/50
D) None of these
Let X be a set containing n elements. Two subsets A and B of X are chosen at random the probability that A B = X is A)
67.
D) 175/1024
One ticket is selected at random from 100 tickets numbered 00, 01, 02, ..., 99. Suppose X and Y are the sum and product of the digit found on the ticket P (X = 7/Y = 0) is given by A) 2/3
66.
C) 45/1024
One mapping is selected at random from all the mappings of the set A = {1, 2, 3, ..., n} into itself. The probability that the mapping selected is one to one is given by A)
65.
B) 7/128
Two players A and B throw a die alternately for a prize of Rs. 11/- which is to be won by a player who first throws a six. If A starts the game, their respective expectations are A) Rs. 6: Rs. 5
64.
D) 27/64
The probability that the 13th day of a randomly chosen month is a second Saturday, is A) 1/7
63.
C) 9/64
The probability of guessing correctly atleast 8 out of 10 answers on a true-false examination, is A) 7/64
62.
B) 9/37
2n
Cn / 2 2n
B) 1 / 2n Cn
C) 1.3.5... (2n – 1)/2nn !
D) (3/4)n
A natural number x is chosen at random from the first 100 natural numbers. The probability that x + 100 50 is x
A) 1/10 68.
D) None of these
B) 55/4098
C) 55/2048
D) None of these
A die is rolled three times, the probability of getting a larger number than the previous number each time is A) 15/216
70.
C) 1/20
If X and Y are independent binomial variates B(5, 1/2) and B(7, 1/2) then P(X + Y = 3) is A) 55/1024
69.
B) 11/50
B) 5/54
C) 13/216
D) 1/18
Let A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8}. An element (a, b) of their cartesian product A X B is chosen at random. The probability that a + b = 9, is A) 1/5
195
B) 2/5
C) 3/5
D) 4/5
IIT-MATHEMATICS-SET-IV 71.
Dialing a telephone number, a man forgot the last two digits and remembering only that they are different, dialed them at random The probability of the number being dialed correctly is A) 1/2
72.
If
1 1 p 3 2
B)
1 2 p 3 3
B) 4/29
B) 4/5
B) 1/5
B) 10
D) None of these
C) 5/29
D) None of these
C) 5/6
D) 1/2
C) 1/3
D) 1/9
C) 12
D) 14
B) E2 and E3 are independent D) E1,E2, E3 are independent
A natural number is selected at random from the set X = {x/1 £ x £ 100}. The probability that the number satisfies the inequation x2 – 13x £ 30, is A) 9/50
80.
1 1 p 6 2
A bag contains four tickets marked with 112, 121, 211, 222 one ticket is drawn at random from the bag. Let E1 { i = 1,2,3}denote the event that ith digit on the ticket is 2. Then A) E1 and E2 are independent C) E3 and E1 are independent
79.
C)
A fair coin is tossed n times. Let X = the number of times head occurs. If P(X = 4), P(X = 5) and P(X = 6) are in A.P., then value of n can be A) 7
78.
D) 1/49
Seven digits from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 are written in a random order. The probability that this seven digits number is divisible by 9 is A) 2/9
77.
C) 1/8
A fair die is thrown until a score of less than five points is obtained. The probability of obtaining less than three points on the last throw is A) 3/4
76.
B) 1/7
Two numbers x and y are chosen at random from the set {1, 2, 3, ...., 30}. The probability that x2 – y2 is divisible by 3 is A) 3/29
75.
D) 1/90
1 4 p 1 p 1 2 p , , are probabilities of three mutually exclusive events, then 4 4 2
A) 74.
C) 1/72
If the integers l and m are chosen at random between 1 to 100 then the probability that a number of the from 7l + 7m is divisible by 5 is A) 1/4
73.
B) 1/45
B) 3/20
C) 2/11
D) None of these
Two integers x and y are chosen, without replacement, at random from the set {x/0£x£10, x is an integer} the probability that |x – y|£ 5 is A) 87/121
B) 89/121
C) 91/121
D) 101/121 196
PROBABILITY 81. If A and B are independent events such that 0 < P(A) <1, 0
82.
A) A, B are mutually exclusive
B) A and are independent
C) A, B are independent
D) P(A/B) + P( A / B ) = 1
A second order determinant is written down at random using the numbers 1, –1 as elements. The probability that the value of the determinant is non zero is A) 1/2
83.
A)
B) 1/2
C) 1/8
C2 x 30 C2 50 C5
30
B)
C2 x 19 C2 50 C5
B) A and B are mutually exclusive D) P(B’/A’) = 1/2
B) p = 2/3, q = 1/2 D) There are infinitely many values of p and q
All the spades are taken out from a pack of cards. From these cards; cards are drawn one by one without replacement till the ace of spades comes. The probability that the ace comes in the 4th draw is B) 12/13
C) 4/13
D) None of these
A die is thrown 2n + 1 times n Î N. The probability that faces with even numbers show odd number of times is A)
197
D) None of these
A student appears for test I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II, III are p, q and 1/2 respectively. If the probability that the student is successful is 1/2 then
A) 1/13 89.
C)
C2 x 31C3 50 C5
For two events A and B, If P(A) = P(A/B) = 1/4 and P(B/A) = 1/2, then
A) p = 1, q = 0 C) p = 3/5, q = 2/3 88.
19
B) P(A B’) 1/3 D) 1/6 P(A’ B’) 1/2
A) A and B are independent C) P(A’/B) = 3/4 87.
D) 1/16
If A and B are two events such that P(A) = 1/2 and P(B) = 2/3, then A) P(AB) 2/3 C) 1/6 P(A B) 1/2
86.
D) 1/3
x1, x2, x3,...x50 are fifty real numbers such that xr < xr+1 for r = 1,2,3 ..., 49. Five numbers out of these are picket up at random. The probability that the five numbers have x20 as the middle number is 20
85.
C) 5/8
If A and B are events at the same experiments with P(A) = 0.2, P(B) = 0.5, then maximum value of P(A’ B) is A) 1/4
84.
B) 3/8
2n 1 2n 3
B) less than
1 2
C) greater than
1 2
D) None of these
IIT-MATHEMATICS-SET-IV 90.
Suppose n boys and m girls take their seats randomly round a circle. The probability of their sitting is
2n 1
1
Cm when
A) no two boys sit together C) boys and girls sit alternatively 91.
B) no two girls sit together D) all the boys sit together
Let A,B,C be three mutually independent events. Consider the two statements S1 and S2. S1 : A and B C are independent S2 : A and B C are independent then A) Both S1 and S2 are true C) Only S2 is true
92.
Sixteen players P1, P2, ..., P3 play in a tournament. They divided into eight pairs at random. From each pair a winner is decided on the basis of a game played between the two players of the pair. Assuming that all the players are of equal strength the probability that exactly one of the two players P1 and P2 is among the eight winners is A) 4/15
93.
1 26
B) 37/50
C) 13/50
D) 27/50
56 36 B) 106
56 2.46 36 C) 10 6
56 46 D) 10 6
B)
56 36 106
C)
56 2.46 36 10 6
D)
56 46 10 6
Two faces of a symmetric die are marked with 1 and remaining faces are marked with 2. A person who speaks truth 2 out of 3 times tosses the die and reports that it is ‘2’ then the chance that it is actually 2 is A) 2/5
97.
D) 17/30
In the above experiment, the chance that the minimum is 3 and maximum is 7 is A)
96.
C) 8/15
A box contains 10 cards numbered 1, 2, 3,....10. 6 cards, are drawn in succession one by one with replacement. Then what is the chance that the maximum of the numbers drawn is not greater than 7 and minimum is not less than 3 1 A) 6 2
95.
B) 7/15
If an integer is selected at random from the first 100 natural numbers then the probability that it is divisible by neither of 2, 3 and 5 is A) 39/50
94.
B) Only S1 is true D) Neither S1 nor S2 is true
B) 3/5
C) 4/5
D) 1/5
The key of a lock is mixed with 9 other keys. If the keys are drawn one by one without replacement and tried the chance that the lock is opened in 6th trial is A) 1/10
B) 1/9
C) 2/3
D) 1/3
198
PROBABILITY 98. 2n equally strong players are participating in a knockout tournament. The players are drawn in pairs at random in each round. Then the probability that two particular players meet in the course of the tournament is A) 99.
1 n
1 2n
B)
1 n 1
C)
D)
1 2 n 1
A bag contains 3 balls of unknown colours. A ball is drawn at random and found to be white. If another ball is drawn without replacing the first ball, the probability that it is white is A) 2/3
B) 1/3
C) 2/5
D) 3/5
100. Probability of Ramesh using car, scooter, bus and train are 1/7, 2/7, 3/7, 1/7 respectively. The probability of him reaching office late with these vehicles are 2/9, 4/9, 1/9, 1/9 respectively. If he reaches office on time, then the probability that he went by car is A) 1/7
B) 2/7
C) 3/7
D) 4/7
101. A box contains 100 tickets numbered 1,2,3,…..100. Two tickets are chosen at random. If it is given that the maximum number on the two chosen tickets is not more than 10, then the probability that the minimum number on them is not less than 5 is : A) 1/3
B) 1/2
C) 152/165
D) 1/9
102. The probability that a man will be alive for 10 more years is 1/4 and that of his wife willlive 10 more years is 1/3. The probability that neither will be alive in 10years is A) 5/12
B) 1/2
C) 7/12
D) 11/12
103. You are given a box with 20 cards in it. 10 of these cards have letter I printed on them and the other ten have the letter T printed them. If you pick up 3 cards (one by one) at random and keep them in same order the probability of making the word I.I.T. is A) 9/80
B) 1/8
C) 4/27
D) 5/38
104. n biscuits are distributed among N boys at random. The probability that a particular boy r(
r
n
B) Cr
1 N 1 C) Cr N N n
n r
1 D) Cr N
gets
r
n
105. One ticket is selected at random from 100 tickets numbered 00,01,02,……,99. Suppose S and T are the sum and product of the digits of the number on the ticket, then the probability of getting S =9 and T=0 is: A) 19/100
B) 1/4
C) 2/19
D) 1/50
106. A natural number x is chosen at random from first 100 natural numbers. Then probability that
199
IIT-MATHEMATICS-SET-IV x
100 50 is x
A) 1/10
B) 11/50
C) 11/20
D) None of these
107. Three letters are written to different persons and addresses on the envelops are also written. Without looking at the address, the letters are put into envelops; the probability that letters go into right envelopes is : A) 1/27
B) 1/6
C) 1/9
D) 1/8
108. A bag contains four tickets with numbers 112, 121, 211, 222. One ticket is drawn at random. Let E1 (i =1,2,3) denote the event that ith digit on the ticket taken out is 2, then : A) E2, E3 are independent C) E1 and E2 are not independent
B) E1, E2, E3 are mutually exclusive D) None of the above
109. A seven digit number is formed by using the digits 1,2,3,4,5,6,7,8,9; no digit being any number. The Probability that the number so formed is a multiple of 9 is : A) 2/9
B) 1/5
C) 1/3
repeat ed in
D) 1/9
110. One hundred identical coins, each with probability p, of showing up heads are tossed once. If 0 < p<1 and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, then the value of p is A) 1/2
B) 49/101
C) 50/101
D) 51/101
111. The probability that two primes and divide a positive integer x is p2 1 A) p p 1 2
1 1 B) p p 1 2
1 1 C) p p 1 2
1 D) p p 1 2
112. In a multiple-choice questions, there are four alternative answers, of which one or more than one are correct. A candidate will get marks for the question only if he ticks all the correct answers. The candidate decides to tick answers at random. He is allowed three chances to answer the questions, then the probability that he will get marks in the question is : A)
2044 3375
B)
631 3375
C)
1 3
D)
1 5
113. Two persons throw a pair of dice alternately till one gets a total of 9 and wins the game. It A has first throw, then the probability that A wins the game is : A) 1/2
B) 7/17
C) 7/11
D) None of these
114. A single letter is selected at random from the word ‘PROBABILITY’ The chance that it is a consonant is:
200
PROBABILITY A) 4/11
B) 1/3
C) 7/11
D) None of these
115. Four digit numbers with distinct digits are formed using the digits 1,2,3,4,5,6,7,8. One number from there is picked up at random. What is the chance that the selected number contains the digit 1? A) 1/2
B) 1/4
C) 1/8
D) None of these
116. A and B are independent witness in a case. The chance that A speaks the truth is x and that B speaks the truth is y. A and B agree on a certain statement, the probability that the statement is true is :
xy xy A) 1 x 1 y xy B) 1 x 1 y
1 x 1 y C) xy 1 x 1 y
D)None of the above
117. A mapping is selected at random from the set of all mappings of the set S = {1,2,3,…..n} to The probability that the selected mapping is onto is A) 118
1 n4
B)
1 n!
C)
n! nn
itself.
D) None of these
A student appears for tests I, II and III. The student is successful if he passes either in test
I and II
or in tests I and III. The probabilities of the students passing in tests I, II and III are p, q and respectively. If the probability that the student is successful is A) p q 1
B) p 0,q 1
1 2
1 then : 2
C) p 1,q 0
D) p=1, q=1/2
119. For any two event A and B associated with an experiment: A) P A. / B
P( A ) P( B ) 1 , where P( B ) 0 P( B )
B) P A B C P( A ) P( A B ) C) P A B 1 P AC P B C if A and B are independent D) P A B 1 P( AC )P( BC ) if A and B are disjoint. 120. A box contains fifty tickets each bearing one of the numbers from 1 to 50. Five tickets are drawn out at random and arranged in ascending order such as x1 x2 x3 x4 x5 . The chance that x3 30 is 30 A) 50
B)
C 29,2 C 20,2 C 50,5
C)
P 29,2 P 20,2 P 50,5
D) none of the above
121. Among 3n consecutive natural numbers, 3 numbers are choosen at random. The probability that the sum of three numbers is a multiple of 3 is 201
IIT-MATHEMATICS-SET-IV 3n 3 3n2 2n A) 3n 1 3n 2
3n 2 3n 2 B) 3n 1 3n 2
3n 3 3n 2 2n D) 3n 1 3n 2
3n 2 C) 3n 1
122. Three of six vertices of a rectangular hexagon are choosen at random. The possibility that angle with three vertices is equilateral equals: A) 1/2
B) 1/5
C) 1/10
the tri-
D) 1/12
123. ‘A’ and ‘B’ are two independent events. If both events ‘A’ and ‘B’ occur together with 1/6 probability. Probability of their not occurring is 1/3. Probability of ‘A’ occurring is : A)
1 1 or 3 4
1 1 B) or 2 3
1 1 C) or 4 2
D) 0 or 1
124. The probability that a student passes in Mathematics, Physics and Chemistry are m, p, and c respectively. Of these subjects, the student has 75% chance of passing in at least one, a 50% chance of passing in at least two and a 40%chance of passing in exactly two. Which of the following relations are true ? A) p m c
19 20
B) p m c
27 20
C) pmc
1 10
D) pmc
1 4
125. 5 persons entered the lift cabin on the ground floor of an 8 floor house. Suppose that each o f them independently and with equal probability can leave the cabin at any floor beginning with the first, then the probability of all 5 persons leaving at different floors is : 75 B) 7 P5
7
P A) 55 7
6 C) 6 P 5
5
D)
P5 55
126. For three events A, B and C, P (exactly one of the events A or B occurs)=P (exactly one of events B or C occurs)=P(exactly one of the events C or A occurs)=p and P(all the three events occur simultaneously) =p2, where 0< p<
1 . Then the probability of at least one of 2
t h e
t h e
three events A, B and C occurring is :
3 p 2 p2 A) 2
p 3 p2 B) 2
3 p p2 C) 2
3 p 2 p2 D) 4
127. A set ‘A’ contains n distinct elements. A subset ‘P’ of A is selected at random. Again a subset Q of A is formed after returning the elements of P. the chance that the two subsets P and Q have no element in common is 3 A) 4
n
1 B) 4
n
1 C) 2
n
D) None of these
128. In a test an examine either guesses or copies or knows the answer to a multiple choice question with 202
PROBABILITY 1 and the probability that he copies the an3 swer is p. the probability that his answer is correct, given that he copied it, is 1/8 If the probabil-
four choices. The probability that he makes a guess is
ity that he know the answer to question, given that he correctly answered is A) 1/2
B) 1/3
C) 1/6
8 then the value of p is 11
D) 1/12
129. A and B take turn in throwing a pair of dice. A wins if he throws a total of 5 before B throws a total 7. If A has the first throw, the probability of his winning the game is A) 9/41
B) 4/11
C) 4/7
D) 3/7
130. A bag A contains n tickets marked 1,2,3,…..n. If two tickets are drawn, then the chance the difference of the numbers on the tickets exceed m < n-1 is A)
n m n m 1 n n 1
B)
n m m n 1 n n 1
C)
n m n m 1 n n 1
that
D) None of the above
131. If n positive integers are taken at random and multiplied together, then the probability the last digit of the product is 2,4,6 or 8 is 4 A) 5
n
B)
3n 2 n 5n
C)
4 n 2n 5n
D) None of these
132. A and B are two events. Odds against A are 2:1 odds in favour of are 3 :1 If then the ordered pair is 5 3 A) , 12 4
2 3 B) , 3 4
1 3 C) , 3 4
D) None of these
133. Three numbers are selected at random without replacement from the set of numbers {1,2,3,….n} the conditional probability that the third number lies between the first two, if the first number is known to be smaller than the second is : A) 1/6
B) 1/3
C) 1/2
D) None of these
134. A bag contains a white and b black balls. Two players A and B alternately draw a ball from the bag, replacing the ball each time after the draw till one of them draws a white ball and wins the game. Abegins the game. If the probability of A winning the game is three times that of B, the ratio a :b is A) 1 :1
B) 1 :2
C) 2 :1
D) none of these
135. Two numbers X and Y are chosen at random (without replacement) from the set{1,2,……5N} The probability that is divisible by 5 is A)
203
N 1 5N 1
4 4 N 1 B) 5 5N 1
17 N 1 C) 5 5N 1
D) none of these
IIT-MATHEMATICS-SET-IV 136. Let A, B,C be three mutually independent events. Consider the two statements S1 and S2, S1 : A and B C are independent S2:A and B C are independent. Then: A) Both S1 and S2 are true C) Only S2 is true
B) Only S1 is true D) Neither S1 nor S2 is true
137. Each of the n urns contains 4 white and 6 black balls. The n 1 th urn contains 5 white
and 5
black balls. Out of the (n+1)urns is chosen an urn at random and two balls are drawn from it without replacement. Both the balls turn out to be black. If the probability that the (n+1)th urn was chosen to draw the ball is A) 10 138. If P( A )
1 then the value of n is 16
B) 11
C) 12
D) 13
3 2 and P B then : 5 3
A) P A B
2 3
B)
4 3 2 9 P A B C) P A / B D) all the above 15 5 5 10
139. A fair coin is tossed four times. If A is the event that a head occurs on each of the first three tosses, B is the even that a tail occurs in the fourth toss and C is the even that exactly two heads occur in the four tosses, then the events: A) A and B are independent C) C and A are independent
B) A,B, C are pair wise independent D) A, B and C are mutually independent
140. If A and B are two events such that P A
A) P A B
2 3
1 B) P A B
1 3
1 2 and P B then 2 3
C)
1 1 P A B 6 2
D)
1 1 P A1 B 6 2
141. A random variable X follows binomial distribution with mean a and variance b, then : A) a>b>0
a2 B) is a positive integer a b
a C) is a positive integer a b
D) None of the above
142. Suppose X is a random variable which takes values 0,1,2,3,…..and P(X =r) = pq r where
0
q = 1-p and r =0,1,2,………………..then : A) p X a q a
B) P X a b / x a p x b
C) P X a b / x a P( x b )
D) None of the above
204
PROBABILITY 143. A random variable X assumes values which are rational numbers of the form
n n 1 and where n1 n
n 1
n n 1 1 =1,2,3,…………..If P x P X , then = n 1 n x 1 A) P x 1 P x 1 B) P x 1 P x 1 3 2 C) P x < P x 1 D) None of the above 2
144. For two events A and B if P A P B
1 and then: 4
A) A and B are mutually exclusive
B) A and B are independent
C)A is a subset of B
A1 3 P D) B 4
145. Either of two persons throw a pair of dice once. The chance that their throws are identical A) 73/648
B) 1/216
C) 575/648
D) None of these
146. Ram and Shyam throw a pair of dice for a prize of Rs. 110 one who throws 7 first wins prize. If Ram has first throw, then mathematical expectation of Ram is A) 6/11
B) Rs.60
C) Rs.50
3! B)
2
5!
C)
5! 3! 3!
46 221
B)
63 221
C)
81 221
letters
D) None of these
148. Two cards are selected at random from a deck of 52 playing cards. The probability that the cards are greater than 2 but less than 9 is : A)
t h e
D) 5/11
147. The letters of the word “MUMMY” are arranged in a row randomly. The chance that the of the two extremes is M is 3! A) 5!
is
D)
bo th
93 221
149. In a certain town, 40% persons have brown hair, 25% have brown eyes, and 15% have bo th If a person selected at random has brown hair, the chance that the person selected at random with brown hair is with brown eyes is A) 3/20
B) 2/3
C) 1/3
D) 3/8
150. Let A be the set of four elements. From the set of all functions from A to A, a function is sen at random. The chance that the selected function is an onto function is A) 3/32
205
B) 29/32
C) 1/64
D) None of these
cho-
IIT-MATHEMATICS-SET-IV 151. Both A and B throw a dice. The chance that B throws a number not less than that thrown is A) 1/2
B) 21/36
C) 15/36
D) None of these
152. From eighty cards numbered 1 to 80, two cards are picked up at random. The probability both the cards have the numbers divisible by 4 is A) 21/316
B) 19/316
by A
C) 1/4
that
D) None of these
153. There are three works in a library; one is of 3 volumes, second is of 4 volumes and the third is of only are volume. They are placed at random on a shelf. What is the chance that the volumes of the same work are placed together? A) 1/40
B) 3/140
C) 9/70
D) None of these
154. The letters of the word “PROBABILITY” are arranged in all possible ways. The chance that two B’s and also two I’s occur together is : A) 1/55
B) 2/55
C) 4/165
D) None of these
1 1 2 155. The probability that the roots of the equation x nx n 0 are real where n N such that n 5 2 2 is
A) 1/5
B) 2/5
C) 3/5
D) 4/5
156. From a bag containing 100 tickets numbered 1,2,3,…..100, one ticket is drawn. If the number on the ticket is x, the chance that x
A) 0
B)
1 2 is : x
99 100
C) 1
D) None of these
157. If the probability that a man aged x years will die with in a year be p, then the chance that out of 5 men A,B,C,D and E, each aged x years, A will die during the year and be the first to die is : A)
1 4 p 1 p 5
1 5 B) 1 1 p 5
C) 5 1 1 p
5
D) None of these
206
PROBABILITY
KEY 1
2
3
4
5
6
7
8
9
10
11
12 13
14
15
D
B
B
B
C
C
C
C
C
D
C
A
C
A
B
16
17
18 19
20
21
22 23
24
25
26
27 28
29
30
A
D
A
D
B
D
B
C
D
C
B
C
B
B
A
31
32
33 34
35
36
37 38
39
40
41
42 43
44
45
C
D
C
C
A
A
A
C
B
B
A
B
B
B
D
46
47
48 49
50
51
52 53
54
55
56
57 58
59
60
AB
A
C
CD BCD ABC AD AC
C
B
B
D
A
B
61
62
63 64
65
66
67 68
69
70
71
72 73
74
75
B
C
A
C
B
D
D
A
B
A
D
C
D
D
76
77
78 79
80
81
82 83
84
85
86
87
88 89
90
D
AD ABC B
207
C
C BCD A
B
D
B ABCDACDABCD A
D
ABC
IIT-MATHEMATICS-SET-IV
91
92
93 94
95
96
97 98
99 100 101 102 103 104 105
A
C
C
C
C
A
A
A
D
A
D
B
D
C
D
106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
C
B
C
A
D
D
A
C
A
C
A
D
C
C
B
121 122 123 124 125 126 127 128 129 130 131 132 133 134 135
B
C
B
BC
A
A
A
B
D
A
C
A
B
C
C
136 137 138 139 140 141 142 143 144 145 146 147 148 149 150
A
A
B
D
C
A
C
B
D
A
B
B
A
D
A
151 152 153 154 155 156 157
A
B
B
B
C
B
B
208
PROBABILITY
SECTION B -MULTIPLE ANSWER TYPE QUESTIONS 1
The probabilities that a student passes in mathematics, physics and chemistry are m, p and c respectively. Of these subjects, a student has a 75% chance of passing in at least one, a 50% chance of passing in at least two, and a 40% chance of passing in exactly two subjects. Which of the following relations are true? A) p + m + c =
2
19 20
B) p + m + c =
27 20
C) pmc =
1 10
D) pmc =
1 4
If E and F are two events with P(E) £ P(F) > 0 then A) occurrence of E Þ occurrence of F B) occurrence of F Þ occurrence of E C) nonoccurrence of E Þ nonoccurrence of F D) none of the above implications hold
3
If A and B are two events such that P(A È B) ³ A) P(A) + P(B) £
4
11 8
B) P(A) + P(B) £
If E and F are the complementary events of the E and F respectively then A) P(E/F) + P( E / F = 1 C) P + P( = 1
5
B) A and are independent D) P(A/B) + P=1
For any two events A and B A) P(AÇB) ³ P(A) + P(B)-1 C) P(AÇB) = P(A) + P(B) - P(AÈB)
209
B) P(A) + P(B) - P(AÇB) D) none of these
If A and B are independent events such that 0 < P(A) < 1, 0 < P(B) < 1 then A) A, B are mutually exclusive C) are independent
9
B) A, B are mutually exclusive D) P(B) £ P(A)
The probability that exactly one of the independent event A and B occurs is equal to A) P(A) + P(B) - 2P(AÇB) C) P() + P() - 2P(Ç)
8
B) A, B are exhaustive D) A, B are independent
Let A and B be two events such that P(AÇB) = 1/3, P(AÈB) = 5/6 and . Then A) A, B are independent C) P(A) = P(B)
7
B) P(E/F) + P( E / F = 1 D) P+P( = 1
Given that x e [0, 1] and y e [0, 1]. Let A be the event of (x, y) satisfying y2 £ x and B be the event of (x, y) satisfying x2 £ y. Then A) P(AÇB) = 1/3 C) A, B are mutually exclusive
6
3 8
3 1 3 and £ P(A ÇB) £ then 4 8 8 7 C) P(A) + P(B) ³ D) none of these 8
B) P(AÇB) ³ P(A) + P(B) D) P(AÇB) = P(A) + P(B) + P(AÈB)
IIT-MATHEMATICS-SET-IV 10.
A coin is tossed repeatedly. A and B call alternately for winning a prize of Rs.30. One who calls correctly first wins the prize. A starts the call. Then the expectation of A) A is Rs.10
11.
B) B is Rs.10
B) P(M) + P(N) - P(M Ç N) D) P(M N) (M N)
For two given events A and B, P(A ÇB) is A) Not less than P(A) + P(B) - 1 C) Equal to P(A) + P(B) - P(A È B)
13.
D) B is Rs.20
If M and N are any two events, the probability that exactly one of them occur is A) P(M) + P(N) - 2P(M Ç N) C) P(M ) P( N) 2P( M N)
12.
C) A is Rs.20
B) Not greater than P(A) + P(B) D) Equal to P(A) + P(B) + P(A È B)
If E and F are independent events such that 0 < P(E) < 1 and 0 < P(F) < 1, then A) E and F are mutually exclusive B) E and F (the complement of the event F) are independent C) E and F are independent D) P(E?F) + P(/F) = 1
14.
For any two events A and B in sample space A) , P(B) ¹ 0, is always true B) = P(A) - P(A Ç B) does not hold C) P(A È B) = 1 - , if A and B are independent D) P(AÈ B) = 1 - , if A and B are disjoint
15.
Let E and F be two independent events. The probability that both E and F happen is 1/12 and the probability that neither E nor F happens is ½. Then A) P(E) = 1/3, P(F) = ¼ C) P(E) = 1/6, P(F) = ½
16.
Let 0 < P(A) < 1, 0 < P(B) < 1 and P(A È B) = P(A) + P(B) = P(A) P(B) A) P(B?A) = P(B) - P(A) C) P(A È B)’) = P(A’) P(B’)
17.
B) P(E) = ½, P(F) = 1/6 D) P(E) = ¼, P(F) = 1/3
B) P(A’ ÈB’) = P(A’) + P(B’) D) P(A/B) = P(A)
If and are the complementary events of events E and F respectively and if 0 < (F) < 1, then A) P(E /F) + P ( / F) = 1B) P(E/F) + P(E / ) = 1 C) P(/ F) + P(E/) = 1
18.
D) P(E/) + P( = 1.
The probabilities that a student passes in mathematics, physics and chemistry are m, p and c respectively. Of these subjects, a student has a 75% chance of passing in at least one, a 50% chance of passing in at least two, and a 40% chance of passing in exactly two subjects. Which of the following relations are true? 210
PROBABILITY A) p + m + c = 19/20 19.
B) 11
C) 7
D) 14
B) m = 50
C) m = 51
D) m = 52
A throws n + 1 coins and B throws n coins. Let P(m. k) be the probability that A throws m heads and B throws x heads where 0 £ m £ n + 1, 0 £ k £ n then A) P(m,k)=
22.
D) pmc = 1/4
Cards are drawn one by one without replacement until two aces are drawn. Let P(m) be the probability that the event occurs in exactly m trials the P(m) must be zero at A) m = 2
21.
C) pmc = 1/10
Let P (n) be the probability of getting n heads when a coin is tossed m times, if P(4), P(5), P(6) are in A.P. then the possible values of m could be A) 10
20.
B) p + m + c = 27/20
B) P(m,k)=
C) P(m k)=½
D)P(m, k)=
If A1, A2 ........, An be any events of the same sample space then A) = 1 B) S P(Ai) £ 1 if A1, A2 ......, An are disjoint C) S P(Ai) ³ 1 if A1, A2 ....., An are exhaustive events D) none of these
23.
One card is missing from a pack of cards. Let A be the event that missing card is a king. Then two cards are drawn and S be event that they are spades then A) P
24.
B) P(S/A) =
B) A and B are independent if n = 6k D) A and B are independent if n = 6k + 2
Let p be the probability that in a pack of cards two kings are adjacent and q be the probability that no two kings are together then A) p = q
211
D) P(A) = P(A/S)
A number is chosen random from the set of integers 1, 2, 3.............n. Let A and B be the events that the number drawn is divisible by 2 and 3 respectively. Then A) A and B are always independent C) A and B are dependent if n = 10
25.
C) P(A/S) =
B) p < q
C) p + q = 1
D) q =
IIT-MATHEMATICS-SET-IV
KEY 1
2
3
4
5
6
BC
D
16
17
CD
AD BC CD
7
8
9
AC AD
A
A
AC BCD AC BC ACD ABCBCD AC AD
18 19
20
21
22 23
24
10
11
12 13
14
15
25
DC BC BC AC BCD CD
212
PROBABILITY
213
