BANSALCLASSES TARGET IIT JEE 2007
MATHEMATICS
STERLING
QUESTION BANK ON
CONIC SECTION (Parabola, Ellipse & Hyperbola)
Time Limit : 4 Sitting Each of 75 Minutes duration approx.
Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct)
Q.1
Two mutually mutually perpendicu perpendicular lar tangents tangents of the parabola parabola y2 = 4ax meet the axis in P1 and P2. If S is the focus of the parabola then
(A) Q.2
1 l (SP1 )
4
l (SP2 )
is equal to
2
(B)
a
1
(C)
a
(D)
a
(C)
x = tan t ;
(D) x = 1 sint ; y = sin
y = se c t
Let Let 'E' 'E' be the the ellip ellipse se
x
+ cos
t 2
9
x2 a
2
y2 b
2
1
(D) ae
+
4
= 1 & 'C' be the circle x 2 + y2 = 9. Let P & Q be the points (1 , 2) and (B) (B) Q lies lies outs outsid idee both oth C & E (D) P lies inside C but outside E.
Let Let S be the the focu focuss of y2 = 4x and a point P is moving on the curve such that it's abscissa is increasing at the rate of 4 units/sec, then the rate of increase of projection of SP on x + y = 1 when P is at (4, 4) is (A)
2
(B) – 1
(C ) –
2 3
(B) 2
(C )
(D) –
2
Eccentric Eccentricity ity of the hyperb hyperbola ola conjug conjugate ate to the hyperb hyperbola ola
(A) Q.7
2
2
2
y
(2, 1) respectively. Then : (A) (A) Q lies lies insid nsidee C but outsid tsidee E (C) P lies inside both C & E
Q.6
t
t
The magnitude magnitude of the gradient gradient of of the tangent tangent at an extremity of latera recta of of the hyperbola hyperbola
2
Q.5
4a
(B) x 2 2 = 2 cos cos t ; y = 4 cos cos2
is equal to (where e is the eccentricity of the hyperbola) (A) be (B) e (C) ab
Q.4
1
Which one of the the following following equations equations represen represented ted parametric parametrically ally,, represents represents equation equation to to a parabolic parabolic prof profile ile ? (A) x = 3 cos t ; y = 4 sin t
Q.3 Q. 3
1
3
x
2
4
y
2
12
3 2
1 is
(D)
4 3
The points points of contact contact Q and and R of tangent tangent from the the point point P (2, 3) 3) on the parabola parabola y 2 = 4x are (A) (9, 6) and (1, 2)
Bansal C lasses
(B) (1, 2) and (4, 4)
(C) (4, 4) and (9, 6)
Q. B. on Parabola, Ellipse, Hyperbola
(D) (9, 6) and (
1 4
, 1)
[2]
Q.8
2
The eccentri eccentricity city of the ellipse ellipse (x – 3) + (y – 4) =
(A) Q.9
2
3
(B)
2
1
y2
(C)
3
The asympt asymptote ote of the hyper hyperbo bola la
x
2
a
2
y
2
b
2
is
9 1
(D)
3 2
1 3
= 1 form with any tangent to the hyperbola hyperbola a triangle whose whose
area is a2tan in magnitude then its eccentricity is : (A) sec (B) cosec (C) sec2
(D) cosec2
Q.10
A tangent tangent is drawn to the parabola parabola y2 = 4x at the point 'P' whose abscissa lies in the interval [1,4]. The maximum possible area of the triangle formed by the tangent at 'P' , ordinate of the point 'P' and the x-axis is equal to (A) 8 (B) 16 (C) 24 (D) 32
Q.11
From an external point P, pair of tangent lines are drawn drawn to the parabola, y 2 = 4x. If 1 & 2 are the
inclinations of these tangents with the axis of x such that, 1 + 2 = , then the locus of P is : 4
(A) x y + 1 = 0 Q.12
The equati equation on
(B) x + y 1 = 0
x2 29 p
+
y2 4 p
(C) x y 1 = 0
(D) x + y + 1 = 0
= 1 (p 4, 29) represents
(A) an ellipse if p is any constant greater than 4. (B) a hyperbola if p is any constant between 4 and 29. (C) a rectangular hyperbola if p is any constant greater than 29. (D) no real curve if p is less than 29. x2
y2
1 with vertices vertice s A and A', tangent drawn at the point P in the first quadrant meets 9 4 the y-axis in Q and the chord A'P meets the y-axis in M. If 'O' is the origin then OQ 2 – MQ2 equals to (A) 9 (B) 13 (C ) 4 (D) 5
Q.13
For For an ellip ellipse se
Q.14
Length Length of the the normal normal chord chord of the parabo parabola, la, y2 = 4x, which makes an angle of (A) 8
Q.15 Q.1 5
(B) 8 2
(C ) 4
4
with the axis of x is:
(D) 4 2
An ellipse and a hyperbola hyperbola have the same centre centre origin, origin, the same foci and the the minor-axis of the one is the same as the conjugate axis of the other. If e , e be their eccentricities respectively, then e 2 e2 1
equals (A) 1 Q.16
(B) 2
2
(C ) 3
1
2
(D) 4
The coordi coordiantes antes of the ends ends of a focal focal chord chord of a parabola parabola y2 = 4ax are (x1, y1) and (x2, y2) then x1x2 + y1y2 has the value equal to (A) 2a2 (B) – 3a 2 (C) – a2 (D) 4a2
Bansal C lasses
Q. B. on Parabola, Ellipse, Hyperbola
[3]
Q.17 Q.17
The The line, line, l x + my + n = 0 will cut the ellipse
x2 a2
+
y2
= 1 in points whose eccentric eccentric angles differ by
b 2
/2 if : (A) a2l 2 + b2n2 = 2 m2 (C) a2l 2 + b2m2 = 2 n2
(B) a2m2 + b2l 2 = 2 n2 (D) a2n2 + b2m2 = 2 l 2
Q.18
Locus Locus of the feet of the the perpendicul perpendiculars ars drawn drawn from from either foci foci on a variable variable tangent tangent to the hyperbola hyperbola 2 2 16y – 9x = 1 is (A) x2 + y2 = 9 (B) x2 + y2 = 1/9 (C) x2 + y2 =7/144 (D) x2 + y2 = 1/16
Q.19
If the normal normal to a para parabol bolaa y2 = 4ax at P meets the curve again in Q and if PQ and the normal at Q makes angles and respectively with the x-axis then tan (tan + tan ) has the value equal to (A) 0
Q.20
(B) – 2
(D) – 1
2
If the normal normal to the parabol parabolaa y2 = 4ax at the point with parameter t1 , cuts the parabola again at the point with parameter t2 , then (A) 2 < t 22 < 8
Q.21
1
(C ) –
(B) 2 < t 22 < 4
(C) t 22 > 4
(D) t 22 > 8
The locus of the point of instruction of the lines 3 x y 4 3 t = 0 & 3 tx + ty 4 3 = 0 (where t is a parameter) is a hyperbola whose eccentricity is (A)
3
(B) 2
(C )
2
(D)
3
4 3 2
Q.22
The equation equation to the locus of the middle point point of the portion portion of the tangent tangent to the ellipse
x
16
2
y
+
9
= 1
included between the co-ordinate axes is the curve : (A) 9x2 + 16y2 = 4 x2y2 (B) 16x2 + 9y2 = 4 x2y2 (C) 3x2 + 4y2 = 4 x2y2 (D) 9x2 + 16y2 = x2y2 Q.23
A parabo parabola la y = ax2 + bx + c crosses the x axis at ( , 0) ( , 0) both to the right of the origin. A circle also passes through these two points. The length of a tangent from the origin to the circle is : (A)
bc a
(B) ac2
(C)
b a
(D)
c a
Q.24
Two parabolas have the same same focus. focus. If their directrices directrices are the the x axis & the y axis respectively, then the slope of their common chord is : (A) ± 1 (B) 4/3 (C) 3/4 (D) none
Q.25
The locus locus of a point point in the the Argand Argand plane plane that moves moves satisfy satisfying ing the equation, equation, z 1 + i z 2 i = 3 (A) is a circle with radius 3 & centre at z = 3/2 (B) is an ellipse with its foci at 1 i and 2 + i and major axis = 3 (C) is a hyperbola with its foci at 1 i and 2 + i and its transverse axis = 3 (D) is none of the above .
Bansal C lasses
Q. B. on Parabola, Ellipse, Hyperbola
[4]
Q.26
A circle has the same centre centre as an ellipse ellipse & passes through the foci foci F1 & F2 of the ellipse, such that the two curves intersect in 4 points. Let 'P' be any one of their point of intersection. If the major axis of the ellipse is 17 & the area of the triangle PF1F2 is 30, then the distance between the foci is : (A) 11 (B) 12 (C) 13 (D) none
Q.27
The straig straight ht line line joining joining any point point P on the parabo parabola la y2 = 4ax to the vertex and perpendicular from the focus to the tangent at P, intersect at R, then the equaiton of the locus of R is (A) x2 + 2y2 – ax = 0 (B) 2x2 + y2 – 2ax = 0 (C) 2x2 + 2y2 – ay = 0 (D) 2x2 + y2 – 2ay = 0
Q.28
A normal normal chord chord of the parabola parabola y2 = 4x subtending a right angle at the vertex makes an acute angle with the x-axis, then equals to (A) arc tan 2
Q.29
(B) arc sec 3
(C) arc cot 2
If the eccentricity eccentricity of the hyperbo hyperbola la x 2 y2 sec2 = 5 is x2 sec2 + y2 = 25, then a value of is : (A) /6 (B) /4 (C) /3
(D) none
3 times the eccentricity of the ellipse
(D)
/2
Q.30
Point 'O' is the centre centre of the ellipse ellipse with major axis AB & minor minor axis CD. Point F is one one focus of of the ellipse. If OF = 6 & the diameter of the inscribed circle of triangle OCF is 2, then the product (AB) (CD) is equal to (A) 65 (B) 52 (C) 78 (D) none
Q.31
Locus Locus of the the feet of the perpendicula perpendiculars rs drawn drawn from from vertex vertex of the parabola parabola y2 = 4ax upon all such chords of the parabola which subtend a right angle at the vertex is (A) x2 + y2 – 4ax = 0 (B) x2 + y2 – 2ax = 0 (C) x2 + y2 + 2ax = 0 (D) x2 + y2 + 4ax = 0
Q.32
For all real real values values of m, the the straight straight line y = mx + (A) 9x2 + 4y2 = 36
Q.33
4 is a tangent to the curve :
(C) 9x2 4y2 = 36
(D) 4x 4x2 9y2 = 36
C is the centre centre of the circle with centre (0, (0, 1) and radius unity unity.. P is is the parabola parabola y = ax2. The set of values of 'a' for which they meet at a point other than the origin, is (A) a > 0
Q.34
(B) 4x2 + 9y2 = 36
9 m2
1 2
(B) a 0,
A tangent tangent having having slope of
(C) x2
4
to the ellipse ellipse
3
18
y2
+
32
1 , 1 4 2
(D)
1 , 2
= 1 intersects the major & minor axes in points A
& B respectively. respectively. If C is the centre of the ellipse then t hen the area of the triangle tri angle ABC is : (A) 12 sq. units (B) 24 sq. units (C) 36 sq. units (D) 48 sq. units
Q.35
The foci foci of the ellipse ellipse (A) 5
Bansal C lasses
x2
y2
16 b (B) 7
2
1 and the hyperbola
x
2
144
y
2
81
1 25
coincide. Then the value of b2 is
(C ) 9
Q. B. on Parabola, Ellipse, Hyperbola
(D) 4
[5]
Q.36
TP & TQ are are tang tangents ents to the the parab parabola ola,, y2 = 4ax at P & Q. If the chord PQ passes through the fixed point ( a, b) then the locus of T is : (A) ay = 2b (x b) (B) bx = 2a (y a) (C) by = 2a (x a) (D) ax = 2b (y b)
Q.37
Through Through the vertex O of the parabola, parabola, y2 = 4ax two chords OP & OQ are drawn and the circles on OP & OQ as diameters intersect in R. If 1, 2 & are the angles made with the axis by the tangents at P & Q on the parabola & by OR then the value of, cot 1 + cot 2 = (A) 2 tan (B) 2 tan tan ( ) (C ) 0 (D) 2 cot
Q.38 Q.3 8
Locus of the middle points points of the parallel chords with gradient gradient m of the rectangular rectangular hyperbola hyperbola xy = c 2 is (A) y + mx = 0 (B) y mx = 0 (C) my x = 0 (D) my + x = 0
Q.39
If the the chord chord throug through h the point point whose whose eccentric eccentric angles angles are & on the ellipse, 2 2 2 2 (x /a ) + (y /b ) = 1 passes through through the focus, then then the value of (1 + e) tan(/2) tan(/2) is (A) e + 1 (B) e 1 (C ) 1 e (D) 0
Q.40
The The given given circle circle x2 + y2 + 2px = 0, p R touches the parabola y2 = 4x externally, then (A) p < 0 (B) p > 0 (C ) 0 < p < 1 (D) p < – 1
Q.41
The locus locus of the the foot foot of the perpendicular perpendicular from the the centre of the hype hyperbola rbola xy = c 2 on a variable tangent is : (A) (x2 y2)2 = 4c2 xy (B) (x2 + y2)2 = 2c2 xy (C) (x2 + y2) = 4x2 xy (D) (x2 + y2)2 = 4c2 xy
Q.42
The tangent tangent at P to a parabo parabola la y2 = 4ax meets the directrix dire ctrix at U and the latus rectum r ectum at V then SUV (where S is the focus) : (A) must be a right triangle (B) must be an equilateral triangle (C) (C) must must be an isos isosce cele less tria triang ngle le (D) (D) must must be a rig right isos isosce cele less tria trian ngle. le.
Q.43
Given the the base of of a triangle triangle and sum of its its sides then then the locus locus of the centre of its incircle incircle is (A) straight line (B) circle (C) ellipse (D) hyperbola
Q.44
P is is a point point on the hyper hyperbola bola
x2 a2
y2 b 2
= 1, N is the foot of of the perpendicular from P on the transverse
axis. The tangent to the hyperbola at P meets the transverse axis at T . If O is the centre of the hyperbola, the OT. ON is equal to : (A) e2 (B) a2 (C) b2 (D)b2/a2 Q.45
Two parabo parabolas las y2 = 4a(x - l 1) and x2 = 4a (y – l 2) always touch one another, the quantities l 1 and l 2 are both variable. variable. Locus Locus of of their their point point of of contact contact has the the equat equation ion (A) xy = a2 (B) xy = 2a2 (C) xy = 4a 2 (D) none
Q.46
If a norma normall to a parabo parabola la y2 = 4ax make an angle with its axis, then it will cut the curve again at an angle (A) tan –1(2 tan)
Bansal C lasses
1 tan 2
(B) tan1
1 tan 2
(C) cot –1
Q. B. on Parabola, Ellipse, Hyperbola
(D) none
[6]
Q.47
Q.48 Q.4 8
If PN is the perpendicular perpendicular from a point point on a rectangu rectangular lar hyperbola hyperbola x 2 y2 = a2 on any of its asymptotes, then the locus of the mid point of PN is : (A) a circle (B) a parabola (C) an ellipse (D) a hyperbola Which one of the following is the the common common tangent to the ellipses,
x2 a2
b2
y2 b 2
=1&
(A) ay = bx +
a4
a 2 b2 b4
(B) by = ax a4
a2 b2 b4
a4
a2 b2 b4
(D) by = ax +
a 2b2 b4
(C) ay = bx
a4
x2 a2
y2 a2
b2
=1?
Q.49
The vertex vertex of a parabola parabola is (2,2) and and the co-ordin co-ordinates ates of its its two extrimities extrimities of the the latus rectum rectum are (–2,0) and (6,0). The equation of the parabola is (A) y2 – 4y + 8x – 12 = 0 (B) x2 + 4x – 8y – 12 = 0 (C) x2 – 4x + 8y – 12 = 0 (D) x2 – 8y – 4x + 20 = 0
Q.50
The equation equation to the the chord chord joining joining two points points (x 1, y1) and (x 2, y2) on the rectangular hyperbola hyperbola xy = c2 is (A) (C)
Q.51
Q.52
x x1
x2 x
y1
y2
+ +
y y1
y2 y
x1
x2
=1
(B)
=1
(D)
x x1
x2 x
y1
y2
+ +
y y1
y2 y
x1
x2
=1 =1
The length of the chord of the parabola parabola y2 = x which is bisected at the point (2, 1) is (A) 2 3
(B) 4 3
(C) 3 2
(D) 2 5
The normal at a variable point P on an ellipse
x2 a2
y2 b 2
= 1 of eccentricity eccentricity e meets meets the axes of the ellipse ellipse
in Q and R then the locus of the mid-point of QR is a conic with an eccentricity e such that : (A) e is independent of e (B) e = 1 (C) e = e (D) e = 1/e Q.53
If the tangents & normals at the extremities of a focal chord of a parabola parabola intersect at (x1, y1) and (x2, y2) respectively, then : (A) x1 = x2 (B) x1 = y2 (C) y1 = y2 (D) x2 = y1
Q.54 Q.54
If P(x1, y1), Q(x2, y 2), R(x3, y3) & S(x 4, y4) are 4 concyclic points on the rectangular hyperbola x y = c2, the co-ordinates of the orthocentre of the triangle PQR are : (A) (x4, y4) (B) (x4, y4) (C) ( x4, y4) (D) ( x4, y4)
Q.55
If the the chord of contact contact of tangents tangents from a point point P to the parabola parabola y2 = 4ax touches the parabola x2 = 4by, the locus of P is : (A) circle (B) parabola (C) ellipse (D) hyperbola
Bansal C lasses
Q. B. on Parabola, Ellipse, Hyperbola
[7]
Q.56
An ellipse is drawn with major and minor axes of of lengths 10 10 and 8 respectively. respectively. Using one focus focus as centre, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. The radius of the circle is (A)
(B) 2
3
(C ) 2 2
(D)
5
Q.57
The latus rectum rectum of a parabola whose whose focal focal chord PSQ PSQ is such that SP = 3 and SQ = 2 is given given by (A) 24/5 (B) 12/5 (C) 6/5 (D) none of these
Q.58
The chord chord PQ PQ of the rectangular rectangular hyperbo hyperbola la xy = a2 meets the axis of x at A ; C is the mid point of PQ & 'O' is the origin. Then the ACO is : (A) equilateral (B) isosceles (C) right angled (D) right isosceles.
Q.59 Q.59
The The circ circle le x2 + y2 = 5 meets the parabola y2 = 4x at P & Q. Then the length PQ is equal to (A) 2
(B) 2 2
(C ) 4
(D) none
Q.60
A common common tangen tangentt to 9x 2 + 16y2 = 144 ; y2 x + 4 = 0 & x2 + y2 12x + 32 = 0 is (A) y = 3 (B) x = 4 (C ) x = 4 (D) y = 3
Q.61
A conic passes through the point (2, 4) and is such that the segment of any of its tangents at any point contained between the co-ordinate axes is bisected at the point of tangency. tangency. Then the foci of the conic are
2
(A) 2 2 , 0 &
2,0
(C) (4, 4) 4) & ( 4, 4)
4
2 2 & 4
2
(B) 2 2 , 2 2 &
2 , 2 2
(D)
2 , 4
2 ,4
Q.62
If two normals normals to a parabol parabolaa y2 = 4ax intersect at right angles then the chord joining their feet passes through a fixed point whose co-ordinates are (A) ( 2a, 0) (B) (a, 0) (C) (2a, 0) (D) none
Q.63 Q.6 3
The equation equation of a straight line passing through the point point (3, 6) and cutting the the curve y = is (A) 4x + y – 18 =0 (B) x + y – 9 = 0 (C) 4x – y – 6 = 0 (D) none
Q.64
Latus Latus rectum of the conic conic satisfy satisfying ing the the differential differential equatio equation, n, x dy + y dx = 0 and and passing passing throug through h the point (2, 8) 8) is (A) 4 2
Q.65
(B) 8
(C ) 8 2
x orthogonally
(D) 16
The area of the rectangle formed formed by the perpendiculars perpendiculars from from the centre centre of the standard ellipse to the tangent and normal at its point whose eccentric angle is /4 is (A)
a
b2 ab a 2 b2
2
Bansal C lasses
a
b 2 (B) 2 a b 2 ab 2
a b (C) ab a b 2
2
2
2
Q. B. on Parabola, Ellipse, Hyperbola
(D)
a2
a
2
b2
b 2 ab
[8]
Q.66
PQ is a normal normal chor chord d of of the the parabola parabola y2 = 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the xaxis in R. Then the length of AR is : (A) equal to the length of the latus rectum (B) equal to the focal distance of the point P (C) equal to twice the focal distance of the point P (D) equal to the distance of the point P from the directrix.
Q.67
If the normal to the rectangular rectangular hyperbo hyperbola la xy = c2 at the point 't' meets the curve again at 't1' then t3 t1 has the value equal to (A) 1 (B) – 1 (C ) 0 (D) none
Q.68
Locus Locus of the point point of intersection of the perpendicular perpendicular tangents tangents of the curve curve 2 y + 4y 6x 2 = 0 is : (A) 2x 1 = 0 (B) 2x + 3 = 0 (C) 2y + 3 = 0 (D) 2x + 5 = 0
Q.69 Q.69
If tan tan
. tan 2 = 1
a 2
b 2
then the chord joining two points 1 & 2 on the ellipse
a right angle at : (A) focus (C) end of the major axis
Q.70
2a 2
x
2
y
= 1 will subtend
2
(B) 2
(B)
p
(C )
11
(D) none
3
a3
p 2
(C)
4a
3
(D)
p 2
p 2 a
The locus locus of of a point point such that two tangents tangents drawn drawn from from it to the parabo parabola la y2 = 4ax are such that the slope of one is double the other is : 9 2
ax
(B) y2 =
9 4
(C) y2 = 9 ax
ax
x2
(D) x2 = 4 ay
y2
1 such that AOB (where 'O' is the origin) is an a 2 b 2 equilateral triangle, then the eccentricity e of the hyperbola satisfies
AB is a double double ordina ordinate te of of the the hype hyperbola rbola
(A) e > Q.74 Q.7 4
b 2
Length Length of of the the focal focal chord of the parabola parabola y2 = 4ax at a distance p from the vertex is :
(A) y2 = Q.73
1 as the centre , a circle is drawn which is tangent to the 9 16 hyperbola with no part of the circle being outside the hyperbola. The radius of the circle is
(A)
Q.72
a2
y2
(B) centre (D) end of the minor axis
With With one focus focus of the hyperbo hyperbola la
(A) less than 2 Q.71
x2
3
(B) 1 < e <
2 3
(C) e =
2
(D) e >
3
2 3
An ellipse is inscribed in a circle and a point within the circle is chosen at random. If the probability that this point lies outside the ellipse is 2/3 then the eccentricity of the ellipse is : (A)
2 2 3
Bansal C lasses
(B)
5 3
(C)
8 9
Q. B. on Parabola, Ellipse, Hyperbola
(D)
2 3
[9]
Q.75
The triangle PQR of of area 'A' is inscribed inscribed in the parabola parabola y2 = 4ax such that the vertex P lies at the vertex of the parabola and the base QR is i s a focal chord. The modulus of the difference of the ordinates of the points Q and R is : (A)
Q.76 Q. 76
A
(B)
2a
a
(C)
2A
a
(D)
If the product of the perpendicular perpendicular distances from any point on the hyperbola hyperbola e=
x2 a
2
4A a
y2 b
2
1 of eccentricity
hyperbola is 3 from its asymptotes is equal to 6, then the length of the transverse axis of the hyperbola
(A) 3 Q.77
A
(B) 6
(C ) 8
(D) 12
The point(s) point(s) on the parabola parabola y2 = 4x which are closest to the circle, x2 + y2 24y + 128 = 0 is/are :
(B) 2 , 2 2
(A) (0, 0)
(C) (4, 4)
(D) none
Q.78 Q.7 8
A point P moves such that the sum of of the angles angles which which the three three normals normals makes with the axis drawn from P on the standard parabola, is constant. Then the locus of P is : (A) a straight line (B) a circle (C) a parabola (D) a line pair
Q.79 Q.79
If x + iy =
i
where i =
1 and and are non zero real parameters then = constant and
= constant, represents two systems of rectangular hyperbola which intersect at an angle of
(A)
(B)
6
(C)
3
(D)
4
2
Q.80
Three normals normals drawn drawn from from any any point to the parabola parabola y2 = 4ax cut the line x = 2a in points whose ordinates are in arithmetical progression. Then the tangents of the angles which the normals make the axis of the parabola are in : (A) A.P. (B) G.P. (C) H.P. (D) none
Q.81
A circle is described described whose whose centre centre is the vertex vertex and whose whose diameter diameter is three-q three-quarters uarters of the latus latus rectum rectum 2 of the parabola y = 4ax. If PQ is the common chord of the circle and the parabola and L1 L2 is the latus rectum, then the area of the trapezium PL1 L2Q is : 2
(A) 3 2 a Q.82
(B)
2
a 2
(C) 4a2
(D)
2 2
2
a2
The tangent tangent to the hyperbola hyperbola xy = c 2 at the point P intersects the x-axis at T and the y-axis at T. The normal to the hyperbola at P intersects the x-axis at N and the y-axis at N . The areas of the triangles PNT and PN'T' are and (A) equal to 1
Q.83 Q.83
2 1
' respectively, then
(B) depends on t
1
1
'
is
(C) depends on c
(D) equal to 2
1 If y = 2 x 3 is a tangent to the parabola y2 = 4a equal to : x , then ' a ' is equal
(A)
22 3
Bansal C lasses
(B) 1
(C )
3
14 3
Q. B. on Parabola, Ellipse, Hyperbola
(D)
14 3
[10]
Q.84
An ellipse having foci at (3, 3) and (– 4, 4) and passing passing through through the the origin has eccentricity equal to (A)
Q.85
3
(B)
7
2
(C)
7
(D)
7
3 5
The The elli ellips psee 4x 4x2 + 9y 9y2 = 36 and the hyperbola 4x2 – y2 = 4 have the same foci and they intersect at right angles then the equation of the circle through the points of intersection of two conics is (A) x2 + y2 = 5
5 (x2 + y2) – 3x – 4y = 0
(B)
5 (x2 + y2) + 3x + 4y = 0
(C) Q.86
5
(D) x 2 + y2 = 25
Tangents angents are are drawn from the point ( 1, 2) on the parabola y 2 = 4 x. The length length , these tangent tangentss will intercept on the line x = 2 is : (A) 6
(B) 6 2
(C) 2 6
(D) none of these
Q.87
The curve curve describes describes parametrical parametrically ly by x = t2 – 2t + 2, y = t2 + 2t + 2 represents (A) straight line (B) pair of straight lines (C) circle (D) parabola
Q.88
At the point of intersection of the rectangular hyperbola hyperbola xy = c 2 and the parabola y2 = 4ax tangents to the rectangular hyperbola and the parabola make an angle and respectively with the axis of of X, then –1 –1 (A) = tan (– 2 tan ) (B) = tan (– 2 tan ) (C)
Q.89
Q.90
=
1 2
tan –1(– tan )
Q.93
2
tan –1(– tan )
Area of of the quadrilateral formed with the the foci foci of the hyperb hyperbola ola (B) 2(a2 + b2)
x2 a
2
y2 b
2
(C) (a2 + b2)
1 and (D)
1 2
x2 a
2
y2 b
2
1 is
(a2 + b2)
A bar of length 20 units moves with its ends on two fixed straight lines at right angles. angles. A point P marked on the bar at a distance of 8 units from one end describes a conic whose eccentricity is (A)
Q.92
1
The tangent tangent and normal normal at at P(t), for all real positive positive t, to the the parabola parabola y2 = 4ax meet the axis of the parabo parabola la in T and G resp respect ective ively ly,, then then the angle angle at which which the tange tangent nt at P to the parabo parabola la is incl incline ined d to the tangent at P to t o the circle passing through the points P, T and G is (A) cot –1t (B) cot –1t2 (C) tan –1t (D) tan –1t2
(A) 4(a2 + b2) Q.91 Q.9 1
(D) =
5 9
(B)
2 3
(C)
4 9
(D)
5 3
In a square square matrix A of order order 3, a i i = mi + i where i = 1, 2, 3 and and mi's are the slopes (in increasing order of their absolute value) of the 3 normals concurrent at the point (9, – 6) to the parabola y 2 = 4x. Rest all other entries of the matrix are one. The value of det. (A) is equal to (A) 37 (B) – 6 (C ) – 4 (D) – 9
Bansal C lasses
(B) 2x + y = 19
(C ) x + y = 9
Q. B. on Parabola, Ellipse, Hyperbola
2
2 is 4 (D) x + 2y = 8
An equation equation for the line that passes through (10, –1) and is perpendicular to y = (A) 4x + y = 39
x
[11]
Direction for Q.94 to Q.97. (4 questions together) together) A quadratic polynomial polynomial y = f = f (x) with absolute term 3 neither touches nor intersects the abscissa axis and is symmetric about the line x = 1. The coefficient of the leading term of the polynomial is unity. A point A(x1, y1) with abscissa x 1 = 1 and a point B(x2, y 2) with ordinate y2 = 11 are given in a cartisian rectangular system of co-ordinates OXY in the first f irst quadrant on the curve y = f = f (x) (x) where 'O' is the origin. Now answer the following questions:
Q.94
Q.95 Q.96 Q.97
Vertex of the quadratic quadratic polynomial polynomial is (A) (1, 1) (B) (2, 3) (C) (1, 2) The scalar product product of the vectors OA and OB is (A) –18 (B) 26 (C) 22
(D) –22
The area bounded bounded by the curve curve y = f (x) (x) and a line y = 3 is (A) 4/3 (B) 5/3 (C) 7/3
(D) 28/3
(D) none
The The grap graph h of y = f (x) (x) represents a parabola whose focus has the co-ordinates (A) (1, 7/4) (B) (1, 5/4) (C) (1, 5/2) (D) (1, 9/4) Direction for Q.98 to Q.66. (3 questions together) together) 2 2 The graph of the conic x – (y – 1) = 1 has one tangent line with positive slope that passes through the origin. the point of tangency being (a, b). Then
Q.98
a The The value value of sin sin –1 is b (A)
Q.99
5
(B)
12
(C)
6
(D)
3
4
Length Length of the latus rectum of the conic is (A) 1
(B)
2
(C ) 2
(D) none
3
(C ) 2
(D) none
Q.100 Eccentricity Eccentricity of the the conic conic is (A)
4
(B)
3
Select the correct correct alternatives : (More than one are correct) correct)
Q.101 Consider a circle with its centre centre lying on the focus of of the parabola, y2 = 2 px such that it touches touches the directrix of the parabola. Then a point of intersection of the circle & the parabola is : (A)
p , p 2
p , p 2
(B)
(C)
p , p 2
p
(D)
2
, p
Q.102 Identify Identify the the statements statements which are True. True. (A) the equation of the director circle of the ellipse, 5x2 + 9y2 = 45 is x2 + y2 = 14. (B) the sum of the focal distances of the point (0, 6) on the ellipse
x2 25
y 2
+
36
= 1 is 10.
(C) the point of intersection of any tangent to a parabola & the perpendicular to it from the focus lies on the tangent at the vertex. (D) P & Q are the points with eccentric angles & + on the ellipse
x2 a2
y2 b 2
= 1, then the area of the
triangle OPQ is independent of .
Bansal C lasses
Q. B. on Parabola, Ellipse, Hyperbola
[12]
Q.103 For the hyper hyperbola bola
x2 9
y2
3
= 1 the incorrect statement is :
(A) the acute angle between its asymptotes is 60º (B) its eccentricity is 4/3 (C) length of the latus rectum is 2 (D) product of the perpendicular distances from any point on the hyperbola hyperbola on its asymptotes is less than the length of its latus rectum . Q.104 The locus of of the t he mi d point of the f ocal ocal radii of a varia vari able bl e point m ovi ng on the parab parabol ola a, y 2 = 4ax is a parabola parabola whose whose (A) Latus rectum is half the latus rectum of the original parabola (B) Vertex Vertex is i s (a/2, 0) (C) Directrix is y-axis (D) Focus has the co-ordinates (a, 0) Q.105 P is is a point on the parabola parabola y2 = 4ax (a > 0) whose vertex is A. PA is produced to meet the directrix in D and M is the foot of the perpendicular from P on the directrix. If a circle is described on MD as a diameter then it intersects the x axis at a point whose co ordinates are : (A) ( 3a, 0) (B) ( a, 0) (C) ( 2a, 0) (D) (a, 0) Q.106 If the circle circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four four points points P(x1, y1), Q(x2, y2), R(x3, y3), S(x4, y4), then (A) x1 + x2 + x3 + x4 = 0 (B) y1 + y2 + y3 + y4 = 0 (C) x1 x2 x3 x4 = c4 (D) y1 y2 y3 y4 = c4
Q.107 Extremities of the latera recta of the ellipses (A) x2 = a(a – y)
(B) x2 = a (a + y)
x2 a2
y2
1 (a > b) having a given major axis 2a lies on b 2 (C) y2 = a(a + x) (D) y2 = a (a – x)
Q.108 Q.108 Let Let y2 = 4ax be a parabola and x2 + y2 + 2 bx = 0 be a circle. If parabola and circle touch each other externally then : (A) a > 0, b > 0 (B) a > 0, b < 0 (C) a < 0, b > 0 (D) a < 0, b < 0 Q.109 The tangent to the hyperbola, hyperbola, x 2 3y2 = 3 at the point
3 , 0 when associated with two asymptotes
constitutes : (A) isosceles triangle
(B) an equilateral triangle
(C) a triangles whose area is 3 sq. units units
(D) a right right isosce isosceles les triang triangle le .
Q.110 Let P, P, Q and R are three three co-normal co-normal points on the parabola parabola y2 = 4ax. Then the correct statement(s) is/are (A) algebraic sum of the slopes s lopes of the normals at P, Q and R vanishes (B) algebraic sum of the ordinates of the points P, P, Q and R vanishes (C) centroid of the triangle PQR lies on the axis of the parabola (D) circle circumscribing the triangle PQR passes through the vertex of the parabola
Bansal C lasses
Q. B. on Parabola, Ellipse, Hyperbola
[13]
Q.11 Q.111 A variable circle is desc ribed to pass through the point (1, 0) and tangent to the curve y = tan (tan 1 x). The locus of the centre of the circle is a parabola whose : (A) length of the latus rectum is 2 2 (B) axis of symmetry has the equation x + y = 1 (C) vertex has the co-ordinates (3/4, 1/4) (D) none of these Q.112 Which of the following equations equations in parametric form can represent a hyperbola, where 't' is a parameter parameter.. (A) x =
a 2
b 1 t 1 & y = 2 t t t
(B)
(C) x = et + et & y = et et
tx a
y
b
+t=0 &
x a
ty
+
b
1 = 0
(D) x2 6 = 2 cos t & y2 + 2 = 4 cos2
t 2
Q.113 The equations equations of the common common tangents tangents to the ellipse, ellipse, x 2 + 4y2 = 8 & the parabola y2 = 4x can be (A) (A) x + 2y + 4 = 0 (B) (B) x – 2y + 4 = 0 (C) (C) 2x + y – 4 = 0 (D) (D) 2x – y + 4 = 0 Q.114 Variable chords of the parabola y2 = 4ax subtend a right angle at the vertex. Then : (A) locus of the feet of the perpendiculars from the vertex on these chords is a circle (B) locus of the middle points of the chords is a parabola (C) variable chords passes through a fixed point on the axis of the parabola (D) none of these
Q.115 Equations Equations of a common tangent tangent to the two hyperbolas hyperbolas (A) y = x + (C) y = x +
a2
a2
(B) y = x
b2
a2
x2
b2
(D)
x
y2
=1&
b 2 a2 a2
y2 a2
x2 b 2
= 1 is :
b2
b2
Q.116 The equation equation of the the tangent tangent to the parabola parabola y = (x 3)2 parallel to the chord joining the points (3, 0) and (4, 1) is : (A) (A) 2 x 2 y + 6 = 0 (B) 2 y 2 x + 6 = 0 (C) (C) 4 y 4 x + 13 = 0 (D) 4 x 4 y = 13 13 Q.117 Let A be the vertex and L the length of the latus rectum of the parabola, y 2 2 y 4 x 7 = 0. The equation of the parabola with A as vertex, 2L the length of the latus rectum and the axis at right angles to that of the given curve is : (A) x2 + 4 x + 8 y 4 = 0 (B) x2 + 4x 8 y + 12 = 0 (C) x2 + 4 x + 8 y + 12 = 0 (D) x2 + 8x 4 y + 8 = 0
Q.118 The differential equation
dx dy
=
3y 2x
represents a family of hyperbolas (except when it represents a pair
of lines) with eccentricity : (A)
3 5
Bansal C lasses
(B)
5 3
(C)
2 5
Q. B. on Parabola, Ellipse, Hyperbola
(D)
5 2
[14]
Q.119 If a number of of ellipse be described having having the same major major axis 2a but a variable minor minor axis then the tangents at the ends of their latera recta pass through fixed points which can be (A) (0, a) (B) (0, 0) (C) (0, – a) (D) (a, a) Q.120 The straight line y + x = 1 touches the parabola : (A) x2 + 4 y = 0 (B) x2 x + y = 0 (C) (C) 4 x2 3 x + y = 0 (D) x2 2x + 2y = 0 Q.121 Circles are drawn on chords of the rectangular hyperbola hyperbola xy = c 2 parallel to the line y = x as diameters. All such circles pass through two fixed points whose co-ordinates are : (A) (c, c) (B) (c, c) (C) ( c, c) (D) ( c, c)
Bansal C lasses
Q. B. on Parabola, Ellipse, Hyperbola
[15]
ANSWER KEY C , B , A 0 2 1 . Q D , C 6 1 1 . Q D , C , A 2 1 1 . Q D , A 8 0 1 . Q , C , B , A 4 0 1 . Q D
C , A 9 1 1 . Q , C , B , A 5 1 1 . Q D C , B 1 1 1 . Q B , A 7 0 1 . Q D , B 3 0 1 . Q
D , B 8 1 1 . Q C , B , A 4 1 1 . Q , C , B , A 0 1 1 . Q D , C , B , A 6 0 1 . Q D D , C , A 2 0 1 . Q
D , A 1 2 1 . Q B , A 7 1 1 . Q B , A 3 1 1 . Q C , B 9 0 1 . Q D , A 5 0 1 . Q B , 0 1 . Q A 1
r o M S r a e n o n a h t e ( : s t l a r o h t t c e l e e r r o c e v i t a n r e t c e r c e ) t c e
A 6 9 . Q B 0 9 . Q C 4 8 . Q A 8 7 . Q A 2 7 . Q C 6 6 . Q 6 . Q C 0 5 . Q C 4 4 . Q B 8 C 2 4 . Q C 6 3 . Q A 0 3 . Q A 4 2 . Q D 8 1 . Q B 2 1 . Q . Q A 6
B 5 9 . Q C 9 8 . Q D 3 8 . Q C 7 7 . Q C 1 7 . Q A 5 6 . Q 5 . Q C 9 5 . Q C 3 4 . Q D 7 D 1 4 . Q B 5 3 . Q B 9 2 . Q D 3 2 . Q C 7 1 . Q C 1 1 . Q . Q C 5
D 0 0 1 . Q C 4 9 . Q A 8 8 . Q C 2 8 . Q B 6 7 . Q B 0 7 . Q C 4 6 . Q 5 . Q B 8 5 . Q C 2 4 . Q B 6 B 0 4 . Q B 4 3 . Q B 8 2 . Q A 2 2 . Q B 6 1 . Q B 0 1 . Q . Q D 4
C 9 9 . Q D 3 9 . Q D 7 8 . Q D 1 8 . Q C 5 7 . Q B 9 6 . Q A 3 6 . Q 5 . Q A 7 5 . Q D 1 4 . Q C 5 . Q B 9 3 D 3 3 . Q B 7 2 . Q B 1 2 . Q B 5 1 . Q A 9 . Q . Q B 3
D 8 9 . Q C 2 9 . Q B 6 8 . Q B 0 8 . Q A 4 7 . Q D 8 6 . Q B 2 6 . Q 5 . Q B 6 5 . Q A 0 4 . Q B 4 A 8 3 . Q D 2 3 . Q C 6 2 . Q D 0 2 . Q B 4 1 . Q B 8 . Q . Q B 2
D 7 9 . Q D 1 9 . Q A 5 8 . Q D 9 7 . Q D 3 7 . Q B 7 6 . Q C 1 6 . Q 5 . Q D 5 4 . Q C 9 4 . Q C 3 . Q A 7 3 A 1 3 . Q D 5 2 . Q B 9 1 . Q C 3 1 . Q B 7 . Q . Q C 1
l O S r o o y n ( : e t l a t c r o h t t c e l e ) t c e r c s i e n v i t a n r e e r c e
Bansal C lasses
Q. B. on Parabola, Ellipse, Hyperbola
[16]
