Question bank on Circle & Straight line There are 125 questions in this question bank. Select the correct alternative : (Only one is correct)
Q.1
Coordinates Coordinates of the centre of the circle which bisects the circumf circumferen erences ces of the circles x2 + y2 = 1 ; x2 + y2 + 2x – 3 = 0 and x2 + y2 + 2y – 3 = 0 is (A) (–1, –1) (B) (3, 3) (C) (2, 2) (D) (– 2, – 2)
Q.2
One side side of of a squa square re is incline inclined d at an acute acute angle angle with the positive x-axis, and one of its extremities is at the origin. If the remaining three vertices of the square lie above the x-axis and the side of a square is 4, then the equation of the diagonal of the square which is not passing through the origin is (A) (cos + sin ) x + (cos – sin ) y = 4 (B) (B) (cos (cos + sin ) x – (cos – sin ) y = 4 (C) (cos – sin ) x + (cos + sin ) y = 4 (D) (D) (cos (cos – sin ) x – (cos + sin ) y = 4 cos 2
Q.3
The line 2x – y + 1 = 0 is tangent tangent to the circle at the point (2, (2, 5) and the centre of of the circles lies lies on x – 2y = 4. The radius of the circle is (A) 3 5
Q.4
(B) 5 3
(C) 2 5
(D) 5 2
Given Given the family family of lines, lines, a (2x (2x + y + 4) + b (x 2y 3) = 0 . Among the lines of the family, family, the number of lines situated at a distance of 10 from the point M (2, 3) is : (A) 0 (B) 1 (C) 2
(D)
Q.5
The coco-ord ordina inate te of the the point point on the circle circle x² x² + y² 12x 4y+ 4y + 30 = 0, which is farthest from the origin are : (A) (9 , 3) (B) (8 , 5) (C) (12 , 4) (D) none
Q.6
The area of triangle formed by the lines lines x + y – 3 = 0 , x – 3y + 9 = 0 and 3x – 2y + 1= 0 (A)
16 7
sq. units
(B)
10 7
sq. units
(C) 4 sq. units
(D) 9 sq. units
Q.7
The number number of commo common n tang tangent ent(s) (s) to the the circle circless x²+ x² + y² + 2x + 8y 23 = 0 and x²+y² 4x 10y 10y + 19 = 0 is : (A) 1 (B) 2 (C) 3 (D) 4
Q.8
The The four four poin points ts whos whosee coordinates are (2, 1), (1, 4), (4, 5), (5, 2) form : (A) (A) a recta rectang ngle le which which is not not a square square (B) (B) a trap trapez eziu ium m whic which h is not not a para paralle llelo logr gram am (C) a square (D) a rhombus which is not a square.
Q.9
From From the the poin pointt A (0, 3) on the the circ circle le x² x² + 4x + (y 3)² = 0 a chord AB is drawn & extended to a point M such such that AM = 2 AB. The equation of the locus of M is : (A) x² + 8x + y² = 0 (B) x² + 8x + (y 3)² = 0 (C) (x 3)² + 8x + y² = 0 (D) x² + 8x + 8y² = 0
Q.10
A ray of light light passing passing through through the point A (1, 2) is reflected reflected at a point B on the x axis and then passes through (5, 3) . Then the equation of AB is : (A) 5x + 4y = 13 (B) 5x 4y = 3 (C) 4x + 5y = 14 (D) 4x 5y = 6
Q.11
Two circles circles of radii 4 cms & 1 cm touch each other externally externally and is the angle contained by their direct common tangents. Then sin = (A)
Q.12
24 25
12
25
(C)
3
(D) none
4
If A & B are the points points ( 3, 4) and (2, 1), then the co ordinates of the point C on AB produced such that AC = 2 BC are : (A) (2, 4)
Q.13
(B)
(B) (3, 7)
(C) (7, 2)
1 , 5 2 2
(D)
The locus of the mid points of the chords chords of of the the circle circle x 2 + y2 ax by = 0 which which subten subtend d a right right angl anglee at
a , b is : 2 2 (B) ax + by = a 2 + b2
(A) ax + by = 0 (C) x2 + y2 ax by +
a 2 b 2 8
=0
(D) x2 + y2 ax by
a 2 b 2 8
= 0
Q.14
The base BC BC of a triangle triangle ABC is bisected bisected at the point point (p, q) and and the equation to the side AB & AC are px + qy = 1 & qx qx + py = 1 . The The equation equation of the the median through through A is : (A) (p 2q) x + (q (q 2p) y + 1 = 0 (B) (p + q) (x + y) y) 2 = 0 (C) (2pq 1) (px + qy 1) = (p2 + q2 1) (qx + py 1) (D) none
Q.15
From (3 , 4) chords chords are drawn drawn to the circle x² + y² 4x = 0 . The locus of the mid mid points points of the chords is : (A) x² + y² 5x 4y + 6 = 0 (B) x² + y² + 5x 4y + 6 = 0 (C) x² + y² 5x + 4y + 6 = 0 (D) x² + y² 5x 4y 6 = 0
Q.16
The The line liness y y1 = m (x x1) ± a 1 m2 are tangents to the same circle . The radius of the circle is : (A) a/2
(B) a
(C) 2a
(D) none
Q.17
The centre of of the smallest circle circle touching the the circles x² + y² 2y 3 = 0 and x² + y² 8x 18y + 93 = 0 is : (A) (3 , 2) (B) (4 , 4) (C) (2 , 7) (D) (2 , 5)
Q.18
If a, b, c are in harmonical progression progression then the line, line, bcx + cay + ab = 0 passes through a fixed point whose coordinates are : (A) (1, 2) (B) ( 1, 2) (C) ( 1, 2) (D) (1, 2)
Q.19
A rhombus is inscribed in the region common to the two circles x 2 + y2 4x 12 = 0 and x2 + y2 + 4x 12 = 0 with two of its vertices vertices on the line joining the the centres of the circles. The area of the rhombous is : (A) 8 3 sq.units
(B) 4 3 sq.units
(C) 16 3 sq.units
(D) none
Q.20
A variable straight line passes through through the points of intersection of the lines, lines, x + 2y = 1 and and 2x y = 1 and meets the coordinate axes in A & B . The locus of the middle point of AB is : (A) x + 3y 10xy = 0 (B) x 3y + 10xy = 0 (C) x + 3y + 10xy = 0 (D) none
Q.21 Q.2 1
In a right right triangle ABC, right right angled at A, on the legAC as diameter, diameter, a semicircle is described. described. The The chord joinin joining g A with with the the poin pointt of of inter interse secti ction on D of the the hy hypote potenu nuse se and and the semic semicir ircle cle,, the then n the the lengt length h AC equa equals ls to AB AD (A)
AB 2
AD 2
(B)
AB AD AB AD
(C)
AB AD
AB AD (D)
AB2
AD2
Q.22 Q.2 2
A variable straight line passes through a fixed point (a, (a, b) intersecting intersecting the coordinates axes at A & B. If 'O' is the origin then the locus of the centroid of the triangle OAB is : (A) bx + ay 3xy = 0 (B) bx + ay 2xy = 0 (C) ax + by 3xy = 0 (D) none
Q.23
The equation of the circle having the lines y2 2y+4x 2xy = 0 as its normals & passing through the point (2 , 1) is : (A) x 2 + y2 2x 4y + 3 = 0 (B) x 2 + y2 2x+4y 5 = 0 (C) x2 + y2 + 2 x + 4 y 13 = 0 (D) none
Q.24
If P = (1, 0) ; Q = ( 1, 0) & R = (2, 0) are three three given points, then the locus of the points S satisfying satisfying the relation relation,, SQ2 + SR 2 = 2SP2 is : (A) a straight line parallel to xaxis xis (B) a circle passing thr through the origin (C) (C) a circle circle with with the the centr centree at the orig origin in (D) (D) a strai straigh ghtt line line para paralle llell to yaxis .
Q.25
If a circle passes through the point (a , b) & cuts the circle x² + y² = K² orthogonally orthogonally,, then the equation of the locus of its centre is : (A) 2ax + 2by (a² + b² + K²) = 0 (B) 2ax + 2by (a² b² + K²) = 0 (C) (C) x² x² + y² 3ax 4by+(a²+b² K²) = 0 (D) x² x² + y² 2ax 3by+ (a² b² K²) = 0
Q.26 Q.26
The coordinates of the orthocentre of the triangle bounded by the lines, 4x 7y+ 7y + 10 = 0; x + y=5 and 7x + 4y = 15 is : (A) (2, 1) (B) ( 1, 2) (C) (1, 2) (D) (1, 2)
Q.27
The distance between between the chords of contact of tangents tangents to the circle ; x2+ y2 + 2gx+2fy+ 2gx+2fy+ c=0 from the origin & the point (g, (g , f) is : (A)
g
2
f
2
(B)
g2 f 2 c 2
(C)
g2 f 2 c 2 g
2
f
2
(D)
g2 f 2 c 2 g2 f 2
Q.28
The equation equation of of the pair pair of bisectors bisectors of the angles angles between between two straight straight lines is, is, 12x2 7xy 12y2 = 0 . If the equation of one line is 2y x = 0 then the equation of the other line is : (A) 41x 38y = 0 (B) 38 38x 41y = 0 (C) (C) 38x + 41y 41y = 0 (D) (D) 41x 41x + 38y 38y = 0
Q.29
The points points A (a, (a , 0), 0) , B (0, (0 , b), b) , C (c (c , 0) & D (0, (0 , d) are such such that ac = bd & a, b, c, d are all non-z non-zer ero. o. Then the points points : (A) form a parallelogram (B) do not lie on a circle (C) form a trapezium (D) are concyclic
Q.30
The line line joining joining two two points points A (2, 0) 0) ; B (3, 1) 1) is rotated rotated about about A in the the anticlock anticlock wise direction direction through through an angle of 15º . The equation of the line in the new position is : (A) x
(B) x 2y 2 = 0 3 y 2 = 0 (B)
(C)
3 x y 2 3 = 0
(D) none
Q.31
The locus of the centers of the circles circles which cut the circles x 2 + y2 + 4x 6y + 9 = 0 and x2 + y2 5x+4y 2 = 0 orthogonally is (A) (A) 9x + 10y 10y 7 = 0 (B) x y + 2 = 0 (C) 9x 10y + 11 = 0 (D) 9x + 10y + 7 = 0
Q.32
Area of the rhombus rhombus bounded bounded by the four lines, ax ± by ± c = 0 is : (A)
Q.33 Q.33
Q.34
c2
(B)
2 ab
2c
2
(C)
ab
4c
2
(D)
ab
ab 4 c2
Given iven A (1, 1) and AB AB is any line through it cutting the x-axis in B. If AC is perpendicular to AB and meets the y-axis in C, then the equation of locus of mid- point P of BC is (A) x + y = 1 (B) x + y = 2 (C) x + y = 2xy (D) 2x + 2y = 1 The locus of the centers of of the circles such that the point point (2 , 3) is the mid point point of the chord chord 5x + 2y = 16 is : (A) 2x 5y + 11 = 0 (B) 2x + 5y 11 = 0 (C) 2x + 5y + 11 = 0 (D) none
Q.35 Q.3 5
A stick of length 10 units rests rests against against the floor floor & a wall wall of a room room . If the the stick begins begins to slide on on the floor floor then the locus of its middle point is : (A) x 2 + y2 = 2.5 (B) x2 + y2 = 25 (C) x2 + y2 = 100 (D) none
Q.36
The locus locus of the mid points points of the chords chords of the circle circle x² + y² + 4x 6y 12 = 0 which subtend an angle angle of
3
radians at its circumference is :
(A) (x 2)² + (y + 3)² = 6.25 (C) (x + 2)² + (y 3)² = 18.75 Q.37
(B) (x + 2)² + (y 3)² = 6.25 (D) (x + 2)² + (y + 3)² = 18.75
Through a given given point point P (a, b) b) a straight line is drawn to meet the axes at Q & R. If the parallelogram parallelogram OQSR is completed then the equation of the locus of S is (given 'O' is the origin) : (A)
a x
+
b y
=1
(B)
a y
+
b x
=1
(C)
a x
+
b y
=2
(D)
a y
+
b x
= 2
Q.38 Q.38
The The poin points ts (x (x1, y1), (x2, y2), (x1, y2) & (x2, y1) are always : (A) collinear (B) concyclic (C) vertices of a square (D) vertices of a rhombus
Q.39 Q. 39
The number of possible possible straight lines , passing through through (2, (2, 3) and forming forming a triangle with with coordinate axes, whose area is 12 sq. units , is (A) one (B) two (C) three (D) four
Q.40
Two mutually mutually perpendicular straight lines through the origin origin from an isosceles triangle with the line 2x + y = 5 . Then the area of the triangle is : (A) 5 (B) 3 (C) 5/2 (D) 1
Q.41
The angle angle at at which which the circles circles (x – 1) 1)2 + y2 = 10 and x2 + (y – 2)2 = 5 intersect is (A)
(B)
6
(C)
4
(D)
3
2
Q.42
A pair of straight straight lines x 2 – 8x + 12 = 0 and y2 – 14y + 45 = 0 are forming a square. Co-ordinates of the centre of the circle inscribed in the square are (A) (3, 6) (B) (4, 7) (C) (4, 8) (D) none
Q.43
The value of 'c' for which the set, {(x, y)x2 + y2 + 2x 1} {(x, y)x y + c 0} contains only one point in common is : (A) ( , 1] [3, ) (B) { 1, 3} (C) { 3} (D) { 1 }
Q.44
Co-ordinates of the orthocentre of the triangle whose vertices vertices are A(0, 0) , B(3, B(3, 4) and C(4, C(4, 0) is (A) (3, 1)
(B) (3, 4)
(C) (3, 3)
(D)
3, 3 4
Q.45
Three lines lines x + 2y + 3 = 0 ; x + 2y – 7 = 0 and 2x – y – 4 = 0 form form the three three sides of two squares. squares. The equation to the fourth side of each square is (A) (A) 2x – y + 14 = 0 & 2x – y + 6 = 0 (B) (B) 2x – y + 14 = 0 & 2x – y – 6 = 0 (C) (C) 2x – y – 14 = 0 & 2x – y – 6 = 0 (D) (D) 2x – y – 14 = 0 & 2x – y + 6 = 0
Q.46
P is a point point (a, b) in the the first quadrant. If the two circles circles which pass through through P and touch both the co-ordinate axes cut at right angles, then : (A) a2 6ab + b2 = 0 (B) a2 + 2ab b2 = 0 (C) a2 4ab + b2 = 0 (D) a2 8ab + b2 = 0
Q.47
If the vertices P and Q of a triangle PQR are given given by (2, 5) and (4, –11) –11) respectively, respectively, and the point point R moves along the line N: 9x + 7y + 4 = 0, then the locus of the centroid of the triangle PQR is a straight line parallel to (A) PQ (B) QR (C) RP (D) N
Q.48
The range of values of 'a' such that the angle (a, 0) to the circle x2 + y2 = 1 satisfies (A) (1, 2)
(B) 1 , 2
2
between the pair of tangents drawn from the point
< < is : (C)
2 , 1
(D)
1 , 2
2 , 1
Q.49
The points points A(a, 0), B(0, B(0, b), b), C(c, C(c, 0) 0) & D(0, d) are such such that ac = bd & a, b, b, c, d are all nonzero. The the points : (A) form a parallelogram (B) do not lie on a circle (C) form a trapezium (D) are concyclic
Q.50 .50
If (, ) is a point on the circle whose centre is on the x -axis and which touches the line x + y = 0 at (2, –2), then the greatest value of is (A) 4 –
2
(B) 6
(C) 4 + 2 2
(D) 4 +
2
Q.51
Distance of of the point (2, 5) from from the line 3x + y + 4 = 0 measured measured parallel parallel to the line 3x 4y + 8 = 0 is (A) 15/2 (B) 9/2 (C) 5 (D) None
Q.52
Three concentric circles of which the biggest is x2 + y2 = 1, have their radii in A.P. If the line y = x + 1 cuts all the circles in real and distinct points. The interval in which the common difference of the A.P. will lie is
1 (A) 0 , 4 Q.53 Q.53
Give Given n
x
y
a b
1 (B) 0 , 2 2
(C)
2 0 , 4
2
(D) none
= 1 and ax + by = 1 are two two variable lines, 'a' and 'b' being the parameters connected by
the relation a2 + b2 = ab. The locus of the point of intersection has the equation (A) x2 + y2 + xy 1 = 0 (B) x2 + y2 – xy + 1 = 0 (C) x2 + y2 + xy + 1 = 0 (D) x2 + y2 – xy – 1 = 0 Q.54
The chord of contact of the tangents tangents drawn drawn from a point point on the circle, x 2 + y2 = a2 to the circle x2 + y2 = b2 touches the circle circle x2 + y2 = c2 then a, b, c are in : (A) A.P. (B) G.P. (C) H.P. (D) A.G.P.
Q.55
A light beam beam emanating emanating from the point A(3, 10) reflects from the line 2x + y - 6 = 0 and then passes through the point B(5, 6) . The equation of the incident and reflected beams are respectively : (A) (A) 4 x 3 y + 18 = 0 & y = 6 (B) x 2 y + 8 = 0 & x = 5 (C) x + 2 y 8 = 0 & y = 6 (D) none of these
Q.56
If the two circle circles, s, x2 + y2 + 2 g1x + 2 f 1y = 0 & x2 + y2 + 2 g2x + 2 f 2y = 0 touch each then: (A) f 1 g1 = f 2 g2
(B)
f 1 g1
=
f 2 g2
(C) f 1 f 2 = g1 g2
(D) none
1 1 1 , p ; Q = , q Q.57 Q.57 If P ; R = H.P. for , r where xk 0, denotes the kth term of an H.P. x x x p r q k N, then :
(A) Ar. ( PQR) =
p 2 q 2 r 2 2
( p q) 2
(q r) 2 ( r p) 2
(B) PQR is a right angled triangle (C) the points P, P, Q, R are collinear (D) none Q.58
Tangents angents are drawn drawn to the circle x 2 + y2 = 1 at the points where it is met by the circles, x2 + y2 ( + 6)x + (8 2 ) y 3 = 0 . being the variable . The locus of the point of intersection of these tangents is : (A) 2x y + 10 = 0 (B) x + 2y 2 y 10 = 0 (C) x 2y + 10 = 0 (D) 2x + y 10 = 0
Q.59 Q.5 9
The acute acute angle angle between two straight straight lines passing through through the the point point M( 6, 8) and the points in which the line segment 2x + y + 10 = 0 enclosed between the co-ordinate co-ordinate axes is divided in the ratio 1 : 2 : 2 in the direction from the point of its intersection with the x axis to the point of intersection with the y axis is : (A) /3 (B) /4 (C) /6 (D) /12
Q.60
B & C are fixed points points having having coordinates (3, 0) and ( 3, 0) respectively . If the vertical angle BAC is 90º, then the locus of o f the centroid of the ABC ABC has the equation : 2 2 2 2 (A) x + y = 1 (B) x + y = 2 (C) 9 (x (x2 + y2) = 1 (D) 9 (x 2 + y2) = 4
Q.61
Chords Chords of the curve curve 4x2 + y2 x + 4y = 0 which subtend a right angle at the origin pass through through a fixed point point whose whose co-ordi co-ordinate natess are are : (A)
Q.62 .62
If
1 , 4 5 5
(B)
1 , 4 5 5
(C)
1 , 4 5 5
(D)
1 , 4 5 5
a , 1 b , 1 c , 1 d , 1 distinct points on a circle of radius radius 4 units then, , , & are four distinct a b c d
abcd is equal to (A) 4
(B) 1/4
(C) 1
(D) 16
Q.63
The pair of straight straight lines x2 4xy + y2 = 0 together with the line x + y + 4 6 = 0 form a triangle which is : (A) rig right angle ngled d but not not isos isosce cele less (B) righ ight iso isosce sceles les (C) scalene (D) equilateral
Q.64
If two chords, chords, each bisected by the x axis can be drawn to the circle, 2 2 (x + y2) 2ax by = 0 (a 0 , b 0) from the point (a, b/2) then : (A) a2 > 8b2 (B) b2 > 2a2 (C) a2 > 2b2 (D) a2 = 2b2
Q.65
If the line y = mx bisects the angle between the lines ax2 + 2h xy + by2 = 0 then m is a root of the quadratic equation : (A) hx2 + (a b)x h = 0 (B) x2 + h (a b) b) x 1 = 0 2 2 (C) (a b) x + hx (a b) = 0 (D) (a b) x hx (a b) = 0
Q.66 Q.6 6
Tangents are are drawn drawn to a unit circle with centre at the origin from each each point on the line 2x + y = 4. Then the equation to the locus of the middle point of the chord of contact is (A) 2 (x 2 + y2) = x + y (B) 2 (x 2 + y2) = x + 2 y (C) 4 (x 2 + y2) = 2x + y (D) none An equilater equilateral al triangl trianglee has each of of its sides of length length 6 cm . If If (x1, y1) ; (x2, y2) & (x3, y3) are its vertices then the value of the determinant,
Q.67
x1 x2 x3
(A) 192 Q.68 Q.6 8
1 1 1
2
is equal to : (B) 243
(C) 486
(D) 972
Two circles circles whose whose radii are are equal to 4 and 8 intersec intersectt at right angles. angles. The length length of their their common common chord chord is (A)
Q.69
y1 y2 y3
16 5
(B) 8
(C) 4 6
(D)
8 5 5
Points A & B are in the first quadrant ; point 'O' 'O' is the the origin . If the slope of of OA is 1, slope of OB OB is 7 and OA = OB, then the slope of AB is : (A) 1/5 (B) 1/4 (C) 1/3 (D) 1/2
Q.70
The common common chord chord of two intersectin intersecting g circles circles c1 & c2 can be seen from their centres at the angles of 90º and 60º respectively . If the distance between their centres is equal to 3 + 1 then the radii of c1 & c2 are :
(A)
3 &3
(B)
2 & 2 2
(C)
2 &2
(D) 2 2 & 4
Q.71
The co-ordinat co-ordinates es of a point P on the line 2x y + 5 = 0 such that PA PB is maximum where A is (4, 2) and B is (2, 4) will be : (A) (11, 27) (B) ( 11, 17) (C) ( 11, 17) (D) (0, 5)
Q.72 Q.7 2
Three circles circles lie on on a plane so so that each of of them externally externally touches the the other two. two. Two Two of them them has radius radius 3, the third having radius unity . If A, B & C are the points of tangency of the circles then the area of the triangle ABC is (A)
9 7
(B)
4
9 7
(C)
8
9 3
(D) none
16
Q.73
Let the co-ordinates of the two points A & B be (1, 2) and (7, 5) respectively. respectively. The line AB is rotated through 45º in anti clockwise direction about the point of trisection of AB which is nearer to B. The equation of the line in new position is : (A) 2x y 6 = 0 (B) x y 1 = 0 (C) 3x y 11 = 0 (D) none of these
Q.74
A pair of tangents are drawn to a unit circle with centre at the origin and these tangents tangents intersect at A enclosing an angle of 60°. The area enclosed by these tangents and the arc of the circle is (A)
2 3
–
6
(B)
3 – 3
(C)
3
–
3
(D)
6
3 1
6
Q.75
The true set of real real value valuess of for which the point P with co-ordinate (, 2) does not lie inside the triangle formed formed by the lines, x y = 0 ; x + y 2 = 0 & x + 3 = 0 is : (A) ( , 2] (B) [0, ] (C) [ 2, 0] (D) ( , 2] [0, ]
Q.76
If the line line x cos + y sin = 2 is the equation of a transverse transverse common common tangent to the the circles
Q.77
x2 + y2 = 4 and x2 + y2 6 3 x 6y + 20 = 0, then the value value of (A) 5/6 (B) 2/3 (C) /3 The graph graph of the functio function, n, cos x cos (x + 2) cos2 (x+1) is : (A ) a straight trai ght l i ne pas passi ng through (0 , sin sin2 1) with slope 2 (B) a straight line passing through (0 , 0) (C) a parabola parabola with with vertex vertex (1, (1 , sin sin2 1) (D) a straight line passing through through the point
Q.78 Q.7 8
is : (D)
/6
, sin2 1 & parallel parallel to the xaxis . 2
A circle circle is drawn drawn with with y-axis y-axis as as a tangent tangent and and its centre centre at the point point which is the reflection reflection of (3, 4) in the line y = x. The equation of the circle is (A) x2 + y2 – 6x – 8y + 16 = 0 (B) x2 + y2 – 8x – 6y + 16 = 0 (C) x2 + y2 – 8x – 6y + 9 = 0 (D) x2 + y2 – 6x – 8y + 9 = 0
Q.79
Let PQR be a right angled isosceles triangle, right right angled at P (2, 1). If the equation of the line QR is 2x + y = 3, then the equation representing the pair of lines PQ and PR is (A) 3x2 3y2 + 8xy + 20x + 10y + 25 = 0 (B) 3x2 3y2 +8xy 20x 10y + 25 = 0 (C) 3x2 3y2 + 8xy+ 10x + 15y + 20 = 0 (D) 3x2 3y2 8xy 10x 15y 20 = 0
Q.80
A circle circle of constan constantt radiu radiuss ' a ' passes passes through through origin origin ' O ' and cuts the axes of co ordinates in points P and Q, then the equation of the locus of the foot of perpendicular from O to PQ is : (A) (x 2 + y2)
x12 y12
(C) (x 2 + y2)2
Q.81 Q.8 1
= 4 a2
x12 y12
(B) (x2 + y2)2
= 4 a2
A is is a point on either either of of two two lines lines y +
(D) (x 2 + y2) 3 x = 2 at a distance of
x12 y12
x12 y12 4
= a2 = a2
units from from their point of intersection. intersection.
3
The co-ordinates of the foot of perpendicular from A on the bisector of the angle between them are (A)
2 3
, 2
(B) (0, 0)
(C)
2 , 2 3
(D) (0, 4)
Q.82
The circle passing through through the distinct points (1, t) , (t, 1) & (t, t) for all values of ' t ' , passes through the point : (A) ( 1, 1) (B) ( 1, 1) (C) (1, 1) (D) (1, 1)
Q.83
In a triangle ABC, side AB has the equation 2 x + 3 y = 29 and the side AC has the the equation , x + 2 y = 16 . If the the mid point of BC BC is (5, 6) then the equation equation of BC BC is : (A) x y = 1 (B) 5 x 2 y = 13 (C) x + y = 11 (D) 3 x 4 y = 9
Q.84
If a circle circle of constant radius 3k passes through the origin origin 'O' 'O' and meets co-ordinate axes at A and B then the locus of the centroid of the triangle OAB is (A) x2 + y2 = (2k)2 (B) x2 + y2 = (3k)2 (C) x2 + y2 = (4k)2 (D) x 2 + y2 = (6k)2
Q.85
The circumce circumcentre ntre of the triangl trianglee forme formed d by by the lines , x y + 2 x + 2 y + 4 = 0 and x + y + 2 = 0 is (A) ( 2, 2) (B) ( 1, 1) (C) (0, 0) (D) ( 1, 2)
Q.86
The locus locus of the mid points points of the chor chords ds of of the the circle circle x2 + y2 2x 4y 11 = 0 which subtend 600 at the centre is (A) x 2 + y2 4x 2y 7 = 0 (B) x2 + y2 + 4x + 2y 7 = 0 (C) x2 + y2 2x 4y 7 = 0 (D) x2 + y2 + 2x + 4y + 7 = 0
Q.87 Q.87
ABC is an isosceles isosceles triangle triangle . If If the co-ordi co-ordinates nates of the base are (1, 3) and ( 2, 7) , then co-ordinates of vertex A can be : (A)
Q.88
12 , 5
(B)
18 , 5
(C)
65 , 5
(D)
7 , 81
Tangents are drawn from (4, 4) to the the circle circle x 2 + y2 2x 2y 7 = 0 to meet the circle at A and B. The length of the chord AB is (A) 2 3
(B) 3 2
(C) 2 6
(D) 6 2
Q.89
The line x + y = p meets the axis of x & y at A & B respectively . A triangle APQ is inscribed in the triangle OAB, O being the origin, with right angle at Q . P and Q lie respectively on OB and AB . If the area of the triangle APQ is 3/8 th of the area of the triangle OAB, then (A) 2
(B) 2/3
(C) 1/3
AQ BQ
is equal to :
(D) 3
Q.90
The equation equation of of the the image image of the circl circlee x2 + y2 + 16x 24y + 183 = 0 by the line mirror 4x + 7y + 13 = 0 is: (A) x 2 + y2 + 32x 4y + 235 = 0 (B) x2 + y2 + 32x + 4y 235 = 0 (C) x2 + y2 + 32x 4y 235 = 0 (D) x2 + y2 + 32x + 4y + 235 = 0
Q.91
If in triang triangle le ABC ABC , A (1, 10) , circumcentre
31 , 23
co-ordinates of mid-point of side opposite to A is : (A) (1, 11/3) (B) (1, 5) (C) (1,
and orthocentre
3)
113 , 43 then the
(D) (1, 6)
Q.92 Q.9 2
Let x & y be the real real numbers numbers satisfying satisfying the equation equation x2 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are M & m respectively, then the numerical value of M m is : (A) 2 (B) 8 (C) 15 (D) none of these
Q.93
If the straight lines lines , ax + amy + 1 = 0 , b x + (m + 1) b y + 1 = 0 and cx + (m + 2)cy 2)cy + 1 = 0, m 0 are concurrent then a, b, b, c are in : (A) A.P. only for m = 1 (B) A.P. for all m (C) G.P. for all m (D) H.P. for all m.
Q.94 Q.9 4
A line meets the co-ordinate co-ordinate axes in A & B. A circle is circumscribed circumscribed about the triangle OAB. If If d1 & d2 are the distances of the tangent to the circle at the origin O from the points A and B respectively, the diameter of the circle is : (A)
2d1 d 2 2
(B)
d1 2d 2 2
(C) d1 + d2
d1d 2
(D) d d 1 2
Q.95 .95
If x1 , y1 are the the roots roots of x2 + 8 x 20 = 0 , x2 , y2 are the roots roots of 4 x2 + 32 32 x 57 = 0 and 2 x3 , y3 are the roots roots of of 9 x + 72 72 x 112 = 0 , then the points, (x1 , y1) , (x2 , y2) & (x 3 , y3) (A) are collinear (B) form an equilateral triangle (C) form form a right right angled angled isosce isosceles les triang triangle le (D) are concyc concyclic lic
Q.96
Two concentric concentric circles circles are such that that the smaller smaller divides divides the larger larger into two regions regions of equal equal area. If If the radius of the smaller circle is 2, 2 , then the length of the tangent tangent from any point ' P ' on the larger circle to the smaller circle is : (A) 1
Q.97
(B)
2
(C) 2
(D) none
Triangle Triangle formed by the lines x + y = 0 , x – y = 0 and l x + my = 1. If l and m vary subject to the condition l 2 + m2 = 1 then the locus of of its circumcentre circumcentre is (A) (x 2 – y2)2 = x2 + y2 (B) (x2 + y2)2 = (x2 – y2) (C) (x2 + y2) = 4x2 y2 (D) (x 2 – y2)2 = (x2 + y2)2
Q.98 Q.98
The equation equation of a line inclined inclined at an angle angle
4
to the axis X, such that the two circles
x2 + y2 = 4, x2 + y2 – 10x – 14y + 65 = 0 intercept equal lengths on it, is (A) (A) 2x – 2y – 3 = 0 (B) (B) 2x – 2y + 3 = 0 (C) (C) x – y + 6 = 0 (D) (D) x – y – 6 = 0 Q.99 Q.99
The coordinates of three points A(4, 0) ; B(2, 1) and and C(3, 1) determine determine the vertices of an equilateral trapezium ABCD ABCD . The coordinates of the vertex D are : (A) (6, 0) (B) ( 3, 0) (C) ( 5, 0) (D) (9, 0)
Q.100 Q.1 00 Tangents are drawn from from any point on the circle x2 + y2 = R 2 to the circle x2 + y2 = r 2. If the line joining the points of intersection of these tangents with the first circle also touch the second, then R equals (A)
2r
(B) 2r
(C)
2r 2 3
(D)
4 r 3 5
Q.101 The image image of the pair of lines represente represented d by ax2 + 2h xy + by2 = 0 by the line mirror y = 0 is (A) ax2 2h xy by2 = 0 (B) bx2 2h xy + ay2 = 0 (C) bx2 + 2h xy + ay2 = 0 (D) ax2 2h xy + by2 = 0 Q.102 Pair of tangents are drawn drawn from every point on the line 3x + 4y = 12 on the circle x 2 + y2 = 4. Their variable chord of contact always passes through a fixed point whose co-ordinates are (A)
4 , 3 3 4
(B)
3 , 3 4 4
(C) (1, 1)
4 3
(D) 1,
Q.103 Q.1 03 The set of values of 'b' for which the origin and the point (1, (1, 1) lie on the same side of the straight line, 2 a x + a by + 1 = 0 a R, b > 0 are : (A) b (2, 4) (B) b (0, 2) (C) b [0, 2] (D) (2, ) Q.104 The equation equation of the circle symmetric symmetric to the circle x2 + y2 – 2x – 4y + 4 = 0 about the line x – y = 3 is (A) x2 + y2 – 10x + 4y + 28 = 0 (B) x2 + y2 + 6x + 8 = 0 (C) x2 + y2 – 14x – 2y + 49 = 0 (D) x2 + y2 + 8x + 2y + 16 = 0 Q.105 Q.1 05 Which one of the following statement is True True ? (A) The lines 2x + 3y + 19 = 0 and 9x + 6y 17 = 0 cut the coordinate coordinate axes in concyclic concyclic points. (B) The circumcentre, orthocentre, incentre and centroid of the triangle formed by the points A(1, 2) , B(4, 6) , C( 2, 1) are colinear . (C) The mid point of the sides of a triangle are (1, 2) , (3, 1) & (5, 5) . The orthocentre of the triangle has the coordinates (3, 1) . (D) Equation of the line pair through the origin and perpendicular to the line pair x y 3 y2 + y 2 x + 10 = 0 is 3 y2 + x y = 0 Q.106 The locus of the centre of a circle which touches externally the circle , x² + y² 6x 6y + 14 = 0 & also touches the y-axis is given by the equation : (A) x² 6x 10y + 14 = 0 (B) x² 10x 6y + 14 = 0 (C) y² 6x 10y + 14 = 0 (D) y² 10x 6y + 14 = 0
Q.107 Q.1 07 The co-ordinates co-ordinates of the vertices P, P, Q, R & S of square PQRS inscribed in the triangle triangle ABC with vertices vertices A (0, 0) , B (3, 0) & C (2, 1) given that two of its vertices P, Q are on the side AB are respectively (A)
1 , 0 , 3 , 0 , 3 , 1 & 1 , 1 4 8 8 8 4 8
(C) (1, 0) ,
3 , 0 , 3 , 1 & 1 , 1 2 3 2 2
(B)
1 , 0 , 3 , 0 , 3 , 1 & 1 , 1 2 4 4 4 2 4
(D)
3 , 0 , 9 , 0 , 9 , 3 & 3 , 3 2 4 4 4 2 4
Q.108 The equation of the locus of the mid mid points points of the chords of the circle 2 4x2 + 4y2 12x + 4y + 1 = 0 that subtend an angle of at its centre is 3 (A) 16(x² + y²) 48x + 16y + 31 = 0 (B) 16(x² + y²) 48x – 16y + 31 = 0 (C) (C) 16(x 16(x²² + y²) + 48x 48x + 16y 16y + 31 = 0 (D) (D) 16(x 16(x²² + y²) + 48x 48x – 16y 16y + 31 = 0 Q.109 The line 2x + 3y 3y = 12 meets meets the x - axis at A and the the yy - axis at B . The line line through through (5, 5) perpendic perpendicular ular to AB AB meets the x - axis, y - axis & the line AB at C, D, E respectively. respectively. If O is the origin, then the area of of the OCEB OCEB is : (A)
20 3
sq. units
(B)
23
sq. units
3
(C)
26 3
sq. units
(D)
5 52 9
sq. units
Q.110 In the xy plane, the segment with end points (3, 8) and (–5, (–5, 2) is the diameter of the circle. The point (k, 10) lies on the circle for (A) no value of k (B) exactly one integral k (C) exacly one non integral k (D) two real values of k Q.111 Q.111 Let A (3, 2) and B (5, 1). ABP is an equilateral triangle is constructed on the side of AB remote from the origin then the orthocentre of triangle ABP is
(A) 4
(C) 4
1 2 1 6
3,
3,
3 2 3 2
3 1 3
3
(B) 4
(D) 4
1 2 1 6
3,
3,
3 2 3 2
3 1 3
3
Q.112 The vertex of a right angle of a right angled triangle lies on the straight line 2x + y – 10 = 0 and the two other vertices, at points (2, –3) and (4, 1) then the area of triangle in sq. units is (A)
10
(B) 3
(C)
33 5
(D) 11
Select the correct alternatives : (More than one are correct)
Q.113 Q.113 Let u ax + by + a 3 b = 0 v bx ay + b 3 a = 0 a, b R be two straight lines. The equation of the bisectors of the angle formed by k 1u k 2v = 0 & k 1u + k 2v = 0 for non zero real k 1 & k 2 are: (A) u = 0 (B) k 2 u + k 1v = 0 (C) k2 u k 1v = 0 (D) v = 0 Q.114 A tangent drawn from the the point (4, 0) to the circle x 2 + y2 = 8 touches it at a point A in the first quadrant. The coordinates of another point B on the circle such su ch that l (AB) = 4 are : (A) (2, 2)
(B) ( 2, 2)
(C)
2
2,0
(D) 0 , 2 2
Q.115 Consider the equation y y1 = m (x x1) . If m & x1 are fixed and different lines are drawn for different values of y1, then : (A) the lines will pass through a fixed point (B) there will be a set of parallel lines (C) all the lines intersect the line x = x1 (D) all the lines will be parallel to the line y = x1. Q.116 A circle circle passes passes through through the points points ( 1, 1) , (0, 6) and (5, 5) . The The point(s) on this circle, the tangent(s) at which is/are parallel to the straight line joining the origin to its centre is/are : (A) (1, 5) (B) (5, 1) (C) ( 5, 1) (D) ( 1, 5) Q.117 If one one vertex of of an equilateral equilateral triangle triangle of side side 'a' lies at the origin origin and and the other other lies on the line x
3 y = 0 then the co-ordinates of the third vertex are :
(B)
(A) (0, a)
3a 2
,
a
2
(C) (0, a)
(D)
3a 2
,
a
2
Q.118 Equation of a line through (7, 4) and touching the circle, x 2 + y2 6x + 4y 3 = 0 is : (A) 5x 12y + 13 = 0 (B) 12x 5y 64 = 0 (C) x 7 = 0 (D) y = 4 Q.119 Three vertices of a triangle triangle are A(4, 3) ; B(1, 1) and C(7, k) . Value(s) of k for which centroid, orthocentre, incentre and circumcentre of the ABC lie on the same straight line is/are : (A) 7 (B) 1 (C) 19/8 (D) none Q.120 Point M moved along the circle (x 4)2 + (y 8)2 = 20 . Then it broke away from it and moving along a tangent to the the circle, circle, cuts the xaxis at the point ( 2, 0) . The coordinates of the point on the circle at which the moving point broke away can be : (A)
3 , 46 5 5
(B)
2 , 44 5 5
(C) (6, 4)
(D) (3, 5)
Q.121 Straight Straight lines 2x + y = 5 and x 2y = 3 intersect at the point A . Points B and C are chosen on these two lines such that AB = AC . Then the equation of a line BC passing through through the point (2, 3) is (A) 3x y 3 = 0 (B) x + 3y 11 = 0 (C) 3x + y 9 = 0 (D) x 3y + 7 = 0 Q.122 The centre(s) centre(s) of the circle(s) passing passing through through the points (0, (0, 0) , (1, 0) and touching touching the circle 2 2 x + y = 9 is/are :
3 , 1 (A) 2 2
1 , 3 (B) 2 2
1 , 21/2 1 , 21/2 (C) 2 (D) 2
Q.123 Q.123 The x co-ordinates of the vertices of a square of unit area are the roots of the equation x2 3x + 2 = 0 and the y co-ordinates of the vertices are the roots of the equation y2 3y + 2 = 0 then the possible vertices of the square is/are : (A) (1, 1), (2, 1), (2, 2), (1, 2) (B) ( 1, 1), ( 2, 1), ( 2, 2), ( 1, 2) (C) (2, 1), (1, 1), (1, 2), (2, 2) (D) ( 2, 1), ( 1, 1), ( 1, 2), ( 2, 2)
7 and touches 3 , Q.124 A circle passes passes through through the point point touches the line pair x2 y2 2x + 1 = 0. The 2 co-ordinates of the centre of the circle are : (A) (4, 0) (B) (5, 0)
(C) (6, 0)
(D) (0, 4)
Q.125 P (x, y) y) moves such that the area of the triangle formed by P, Q (a , 2 a) and R ( a, 2a) 2 a) is equal to the area of the triangle formed by P, S (a, 2 a) & T (2 a, 3 a). The locus of 'P' is a straight line given by : (A) 3x y = a (B) 5x 3y + a = 0 (C) y = 2ax (D) 2y = ax
ANSWER ANSWE R KEY B , A 5 2 1 . Q
C , A 4 2 1 . Q
B , A 3 2 1 . Q
D , C 2 2 1 . Q
B , A 1 2 1 . Q
C , B 0 2 1 . Q
C , B 9 1 1 . Q
C , A 8 1 1 . Q
, B , A 7 1 1 . Q D , C
D , B 6 1 1 . Q
C , B 5 1 1 . Q
B , A 4 1 1 . Q
D , A 3 1 1 . Q
1 1 . Q B 2
1 1 . Q D 1
B 0 1 1 . Q
B 9 0 1 . Q
A 8 0 1 . Q
D 7 0 1 . Q
D 6 0 1 . Q
A 5 0 1 . Q
A 4 0 1 . Q
B 3 0 1 . Q
D 2 0 1 . Q
D 1 0 1 . Q
0 1 . Q B 0
9 . Q D 9
9 . Q A 8
9 . Q A 7
9 . Q C 6
9 . Q A 5
. Q C 4 9
. Q D 3 9
B 2 9 . Q
9 . Q A 1
D 0 9 . Q
D 9 8 . Q
B 8 8 . Q
D 7 8 . Q
C 6 8 . Q
B 5 8 . Q
A 4 8 . Q
C 3 8 . Q
D 2 8 . Q
B 1 8 . Q
8 . Q C 0
7 . Q B 9
7 . Q C 8
7 . Q D 7
7 . Q D 6
D 5 7 . Q
B 4 7 . Q
C 3 7 . Q
C 2 7 . Q
B 1 7 . Q
C 0 7 . Q
D 9 6 . Q
A 8 6 . Q
D 7 6 . Q
C 6 6 . Q
6 . Q A 5
6 . Q C 4
6 . Q D 3
6 . Q C 2
6 . Q A 1
A 0 6 . Q
B 9 5 . Q
A 8 5 . Q
C 7 5 . Q
B 6 5 . Q
A 5 5 . Q
B 4 5 . Q
A 3 5 . Q
C 2 5 . Q
C 1 5 . Q
C 0 5 . Q
4 . Q D 9
4 . Q D 8
D 7 4 . Q
C 6 4 . Q
4 . Q D 5
4 . Q D 4
4 . Q D 3
4 . Q B 2
4 . Q B 1
A 0 4 . Q
C 9 3 . Q
B 8 3 . Q
A 7 3 . Q
B 6 3 . Q
B 5 3 . Q
A 4 3 . Q
A 3 3 . Q
B 2 3 . Q
C 1 3 . Q
3 . Q C 0
2 . Q D 9
2 . Q A 8
2 . Q C 7
2 . Q C 6
A 5 2 . Q
D 4 2 . Q
A 3 2 . Q
A 2 2 . Q
D 1 2 . Q
A 0 2 . Q
A 9 1 . Q
D 8 1 . Q
D 7 1 . Q
B 6 1 . Q
1 . Q A 5
1 . Q C 4
1 . Q C 3
1 . Q C 2
1 . Q A 1
. Q A 0 1
B 9 . Q
. Q C 8
. Q C 7
. Q B 6
A 5 . Q
B 4 . Q
A 3 . Q
C 2 . Q
D 1 . Q
