Conic Sections 729

2. PQ is a double ordinate of the parabola y 2 = 4 ax . The locus 14. The ends of latus rectum of parabola x 2 + 8 y = 0 are
of the points of trisection of PQ is [MP PET 1995]
(a) 9 y = 4 ax
2
(b) 9 x 2
= 4 ay (a) (–4, –2) and (4, 2) (b) (4, –2) and (–4, 2)
(c) (–4, –2) and (4, –2) (d) (4, 2) and (–4, 2)
(c) 9 y 2 + 4 ax = 0 (d) 9 x 2 + 4 ay = 0
15. The end points of latus rectum of the parabola x 2 = 4 ay are
3. If the vertex of a parabola be at origin and directrix be [RPET 1997]
x + 5 = 0 , then its latus rectum is [RPET 1991] (a) (a, 2 a), (2 a, − a) (b) (−a, 2 a), (2 a, a)
(a) 5 (b) 10
(c) (a, − 2 a), (2 a, a) (d) (−2 a, a), (2 a, a)
(c) 20 (d) 40
4. The latus rectum of a parabola whose directrix is 16. The equation of the parabola with its vertex at the origin, axis
on the y-axis and passing through the point (6, –3) is
x + y − 2 = 0 and focus is (3, – 4), is
[MP PET 2001]
(a) − 3 2 (b) 3 2 (a) y = 12 x + 6
2
(b) x 2
= 12 y
(c) − 3 / 2 (d) 3 / 2 (c) x 2
= −12 y (d) y = −12 x + 6
2

5. The equation of the lines joining the vertex of the parabola 17. Focus and directrix of the parabola x 2 = −8 ay are
y 2 = 6 x to the points on it whose abscissa is 24, is
[RPET 2001]
(a) y  2 x = 0 (b) 2 y  x = 0 (a) (0, − 2 a) and y = 2 a (b) (0, 2 a) and y = −2 a
(c) x  2 y = 0 (d) 2 x  y = 0 (c) (2 a, 0 ) and x = −2 a (d) (−2 a, 0 ) and x = 2 a
6. The points on the parabola y = 36 x whose ordinate is
2
18. The equation of the parabola with focus (3, 0) and the
three times the abscissa are directirx x + 3 = 0 is [EAMCET 2002]
(a) (0, 0), (4, 12) (b) (1, 3), (4, 12) (a) y 2 = 3 x (b) y 2 = 2 x
(c) (4, 12) (d) None of these (c) y 2 = 12 x (d) y 2 = 6 x
7. The points on the parabola y = 12 x whose focal distance is
2
19. Locus of the poles of focal chords of a parabola is of parabola
4, are [EAMCET 2002]
(a) The tangent at the vertex (b) The axis
(a) (2, 3 ), (2, − 3 ) (b) (1, 2 3 ), (1,−2 3 )
(c) A focal chord (d) The directrix
(c) (1, 2) (d) None of these
20. The parabola y 2 = x is symmetric about
8. The focal distance of a point on the parabola y 2 = 16 x [Kerala (Engg.) 2002]
whose ordinate is twice the abscissa, is (a) x-axis (b) y-axis
(a) 6 (b) 8 (c) Both x-axis and y-axis (d) The line y = x
(c) 10 (d) 12
21. The point on the parabola y 2 = 18 x , for which the ordinate
9. The co-ordinates of the extremities of the latus rectum of the
is three times the abscissa, is [MP PET 2003]
parabola 5 y 2 = 4 x are
(a) (6, 2) (b) (–2, –6)
(a) (1 / 5, 2 / 5 ), (−1 / 5, 2 / 5 ) (b) (1 / 5, 2 / 5 ), (1 / 5, − 2 / 5 ) (c) (3, 18) (d) (2, 6)
(c) (1 / 5, 4 / 5 ), (1 / 5, − 4 / 5 ) (d) None of these 22. The equation of latus rectum of a parabola is x + y = 8 and
the equation of the tangent at the vertex is x + y = 12 , then
10. A parabola passing through the point (−4 , − 2) has its vertex
length of the latus rectum is [MP PET 2002]
at the origin and y-axis as its axis. The latus rectum of the
parabola is (a) 4 2 (b) 2 2
(a) 6 (b) 8 (c) 8 (d) 8 2
(c) 10 (d) 12
23. Vertex of the parabola y + 2 y + x = 0 lies in the quadrant
2

11. The focus of the parabola x 2 = −16 y is [MP PET 1989]

[RPET 1987; MP PET 1988, 92] (a) First (b) Second
(a) (4, 0) (b) (0, 4) (c) Third (d) Fourth
(c) (–4, 0) (d) (0, –4) 24. The equation x 2 − 2 xy + y 2 + 3 x + 2 = 0 represents
12. If (2, 0) is the vertex and y-axis the directrix of a parabola, [UPSEAT 2001]
then its focus is [MNR 1981] (a) A parabola (b) An ellipse
(a) (2, 0) (b) (–2, 0) (c) A hyperbola (d) A circle
(c) (4, 0) (d) (–4, 0) 25. x − 2 = t 2 , y = 2 t are the parametric equations of the
13. If the parabola y = 4 ax passes through (–3, 2), then length
2
parabola
of its latus rectum is [RPET 1986, 95]
(a) y 2 = 4 x (b) y 2 = −4 x
(a) 2/3 (b) 1/3
(c) x 2 = −4 y (d) y 2 = 4 (x − 2)
(c) 4/3 (d) 4
 730 Conic Sections
26. The equation of the latus rectum of the parabola 37. The equation of the latus rectum of the parabola represented
x 2 + 4 x + 2 y = 0 is [Pb. CET 2004] by equation y 2 + 2 Ax + 2 By + C = 0 is
(a) 2 y + 3 = 0 (b) 3 y = 2 B2 + A2 − C B2 − A2 + C
(a) x = (b) x =
(c) 2 y = 3 (d) 3 y + 2 = 0 2A 2A
B2 − A2 − C A2 − B2 − C
27. Vertex of the parabola 9 x − 6 x + 36 y + 9 = 0 is
2
(c) x = (d) x =
2A 2A
(a) (1 / 3, − 2 / 9 ) (b) (−1 / 3, − 1 / 2)
38. The parametric equation of the curve y 2 = 8 x are
(c) (−1 / 3, 1 / 2) (d) (1 / 3, 1 / 2)
(a) x = t 2 , y = 2 t (b) x = 2 t 2 , y = 4 t
28. The equation of the parabola whose axis is vertical and
passes through the points (0, 0), (3, 0) and (–1, 4) is (c) x = 2 t, y = 4 t 2 (d) None of these

(a) x 2 − 3 x − y = 0 (b) x 2 + 3 x + y = 0 t t2
39. The equations x = ,y= represents
4 4
(c) x 2 − 4 x + 2 y = 0 (d) x 2 − 4 x − 2 y = 0
(a) A circle (b) A parabola
29. The equation of the parabola whose vertex is (–1, –2), axis (c) An ellipse (d) A hyperbola
is vertical and which passes through the point (3, 6), is
40. The equation of parabola whose vertex and focus are (0, 4)
(a) x 2 + 2 x − 2 y − 3 = 0 (b) 2 x 2 = 3 y and (0, 2) respectively, is [RPET 1987, 89, 90, 91]

(c) x 2 − 2 x − y + 3 = 0 (d) None of these (a) y 2 − 8 x = 32 (b) y 2 + 8 x = 32

(c) x 2 + 8 y = 32 (d) x 2 − 8 y = 32
30. Axis of the parabola x 2 − 4 x − 3 y + 10 = 0 is
41. Curve 16 x 2 + 8 xy + y 2 − 74 x − 78 y + 212 = 0 represents
(a) y + 2 = 0 (b) x + 2 = 0
(a) Parabola (b) Hyperbola
(c) y − 2 = 0 (d) x − 2 = 0
(c) Ellipse (d) None of these
31. Equation of the parabola whose directrix is y = 2 x − 9 and 42. The length of the latus rectum of the parabola
focus (–8, –2) is 9 x 2 − 6 x + 36 y + 19 = 0 [MP PET 1994]
(a) x 2 + 4 y 2 + 4 xy + 16 x + 2 y + 259 = 0 (a) 36 (b) 9
(c) 6 (d) 4
(b) x 2 + 4 y 2 + 4 xy + 116 x + 2 y + 259 = 0
43. The axis of the parabola 9 y 2 − 16 x − 12 y − 57 = 0 is
(c) x + y + 4 xy + 116 x + 2 y + 259 = 0
2 2
[MNR 1995]
(d) None of these (a) 3 y = 2 (b) x + 3 y = 3
32. The equation of the parabola with (–3, 0) as focus and (c) 2 x = 3 (d) y = 3
x + 5 = 0 as directirx, is 44. The vertex of a parabola is the point (a, b) and latus rectum
[RPET 1985, 86, 89; MP PET 1991] is of length l. If the axis of the parabola is along the positive
(a) x 2 = 4 (y + 4 ) (b) x 2 = 4 (y − 4 ) direction of y-axis, then its equation is
l l
(c) y 2 = 4 (x + 4 ) (d) y 2 = 4 (x − 4 ) (a) (x + a) 2 = (2 y − 2b) (b) (x − a) 2 = (2 y − 2b)
2 2
33. The equation of the parabola whose vertex and focus lies on l l
the x-axis at distance a and a’ from the origin, is (c) (x + a) 2 = (2 y − 2b) (d) (x − a) 2 = (2 y − 2b)
4 8
[RPET 2000]
45. If the vertex of the parabola y = x 2 − 8 x + c lies on x-axis,
(a) y = 4 (a'−a)(x − a)
2
(b) y = 4 (a'−a)(x + a)
2
then the value of c is
(c) y 2 = 4 (a'+ a)(x − a) (d) y 2 = 4 (a'+ a)(x + a) (a) –16 (b) –4
(c) 4 (d) 16
34. The focus of the parabola y 2 = 4 y − 4 x is [MP PET 1991]
46. The points of intersection of the curves whose parametric
(a) (0, 2) (b) (1, 2) 2
equations are x = t 2 + 1, y = 2 t and x = 2 s, y = is given
(c) (2, 0) (d) (2, 1) s
by
35. Vertex of the parabola x + 4 x + 2 y − 7 = 0 is
2
(a) (1, − 3) (b) (2, 2)
[MP PET 1990]
(c) (–2, 4) (d) (1, 2)
(a) (–2, 11/2) (b) (–2, 2)
(c) (–2, 11) (d) (2, 11) 47. The latus rectum of the parabola y 2 = 5 x + 4 y + 1 is
[MP PET 1996]
36. If the axis of a parabola is horizontal and it passes through
the points (0, 0), (0, –1) and (6, 1), then its equation is 5
(a) (b) 10
4
(a) y + 3 y − x − 4 = 0
2
(b) y − 3 y + x − 4 = 0
2
5
(c) y − 3 y − x − 4 = 0
2
(d) None of these (c) 5 (d)
2
 Conic Sections 731

48. The equation of the locus of a point which moves so as to be 59. The focus of the parabola y = 2 x 2 + x is [MP PET 2000]
at equal distances from the point (a, 0) and the y-axis is
1 1
(a) y 2 − 2 ax + a 2 = 0 (b) y 2 + 2 ax + a 2 = 0 (a) (0, 0) (b)  , 
2 4
(c) x 2 − 2 ay + a 2 = 0 (d) x 2 + 2 ay + a 2 = 0
 1   1 1
(c)  − , 0  (d)  − , 
49. The focus of the parabola x 2 = 2 x + 2 y is  4   4 8
 3 −1   −1  60. The focus of the parabola y 2 − x − 2 y + 2 = 0 is
(a)  ,  (b)  1, 
2 2   2  [UPSEAT 2000]
(c) (1, 0) (d) (0, 1) (a) (1 / 4 , 0 ) (b) (1, 2)
50. Latus rectum of the parabola y 2 − 4 y − 2 x − 8 = 0 is (c) (3/4, 1) (d) (5/4, 1)
(a) 2 (b) 4 61. The vertex of parabola (y − 2) = 16 (x − 1) is
2

(c) 8 (d) 1 [Karnataka CET 2001]

51. The equation of the parabola with focus (a, b) and directrix (a) (2, 1) (b) (1, –2)
x y (c) (–1, 2) (d) (1, 2)
+ = 1 is given by [MP PET 1997]
a b 62. Equation of the parabola with its vertex at (1, 1) and focus (3,
1) is [Karnataka CET 2001, 02]
(a) (ax − by ) − 2 a x − 2b y + a + a b + b
2 3 3 4 2 2 4
=0
(a) (x − 1) 2 = 8 (y − 1) (b) (y − 1) 2 = 8 (x − 3)
(b) (ax + by ) − 2 a x − 2b y − a + a b − b
2 3 3 4 2 2 4
=0
(c) (y − 1) 2 = 8 (x − 1) (d) (x − 3) 2 = 8 (y − 1)
(c) (ax − by )2 + a 4 + b 4 − 2 a 3 x = 0
63. The equation of parabola whose focus is (5, 3) and directrix
(d) (ax − by ) 2 − 2 a 3 x = 0 is 3 x − 4 y + 1 = 0 , is [MP PET 2002]
52. The length of latus rectum of the parabola (a) (4 x + 3 y ) 2 − 256 x − 142 y + 849 = 0
4 y 2 + 2 x − 20 y + 17 = 0 is [MP PET 1999]
(b) (4 x − 3 y ) 2 − 256 x − 142 y + 849 = 0
(a) 3 (b) 6
(c) (3 x + 4 y ) 2 − 142 x − 256 y + 849 = 0
1
(c) (d) 9
2 (d) (3 x − 4 y ) 2 − 256 x − 142 y + 849 = 0
53. Eccentricity of the parabola x 2 − 4 x − 4 y + 4 = 0 is 64. Which of the following points lie on the parabola x 2 = 4 ay
[RPET 1996; Pb. CET 2003] [RPET 2002]
(a) e = 0 (b) e = 1 (a) x = at , y = 2 at
2
(b) x = 2 at, y = at
(c) e  4 (d) e = 4
(c) x = 2 at , y = at
2
(d) x = 2 at, y = at 2
54. The vertex of the parabola 3 x − 2 y − 4 y + 7 = 0 is
2
65. The equation of the parabola whose vertex is at (2, –1) and
[RPET 1996] focus at (2, –3) is [Kerala (Engg.) 2002]
(a) (3, 1) (b) (–3, –1)
(a) x 2 + 4 x − 8 y − 12 = 0 (b) x 2 − 4 x + 8 y + 12 = 0
(c) (–3, 1) (d) None of these
(c) x 2 + 8 y = 12 (d) x 2 − 4 x + 12 = 0
55. The focus of the parabola 4 y 2 − 6 x − 4 y = 5 is
[RPET 1997] 66. The directrix of the parabola x 2 − 4 x − 8 y + 12 = 0 is
(a) (–8/5, 2) (b) (–5/8, 1/2) [Karnataka CET 2003]

(c) (1/2, 5/8) (d) (5/8, –1/2) (a) x = 1 (b) y = 0

56. The vertex of the parabola x 2 + 8 x + 12 y + 4 = 0 is (c) x = −1 (d) y = −1

[DCE 1999] 67. The equation of the parabola with focus (0, 0) and directrix
x + y = 4 is [EAMCET 2003]
(a) (–4, 1) (b) (4, –1)
(c) (–4, –1) (d) (4, 1) (a) x 2 + y 2 − 2 xy + 8 x + 8 y − 16 = 0
57. Focus of the parabola (y − 2) 2 = 20 (x + 3) is (b) x 2 + y 2 − 2 xy + 8 x + 8 y = 0
[Karnataka CET 1999] (c) x 2 + y 2 + 8 x + 8 y − 16 = 0
(a) (3, –2) (b) (2, –3)
(d) x 2 − y 2 + 8 x + 8 y − 16 = 0
(c) (2, 2) (d) (3, 3)
58. The length of the latus rectum of the parabola 68. If (0, 6) and (0, 3) are respectively the vertex and focus of a
parabola, then its equation is [Karnataka CET 2004]
x 2 − 4 x − 8 y + 12 = 0 is [MP PET 2000]
(a) x 2 + 12 y = 72 (b) x 2 − 12 y = 72
(a) 4 (b) 6
(c) 8 (d) 10 (c) y 2 − 12 x = 72 (d) y 2 + 12 x = 72
 732 Conic Sections
69. The equation of the directrix of the parabola 80. The point of the contact of the tangent to the parabola
x 2 + 8 y − 2 x = 7 is [UPSEAT 2004] y 2 = 4 ax which makes an angle of 60 o with x-axis, is
(a) y = 3 (b) y = −3  a 2a   2a a 
(a)  ,  (b)  , 
(c) y = 2 (d) y = 0
3 3   3 3
70. The equation of axis of the parabola 2 x 2 + 5 y − 3 x + 4 = 0
 a 2a 
is [Pb. CET 2000] (c)  , 
 (d) None of these
3 3  3 3 
(a) x = (b) y =
4 4 81. The straight line y = 2 x +  does not meet the parabola
1 y 2 = 2 x , if [MP PET 1993; MNR 1977]
(c) x = − (d) x − 3 y = 5
2 1 1
(a)   (b)  
71. If x + 6 x + 20 y − 51 = 0 , then axis of parabola is
2 4 4
[Orissa JEE 2004] (c)  = 4 (d)  = 1
(a) x + 3 = 0 (b) x − 3 = 0 82. The equation of the tangent at a point P(t) where ‘t’ is any
(c) x = 1 (d) x + 1 = 0 parameter to the parabola y 2 = 4 ax , is [MNR 1983]
72. The equation of the tangent to the parabola y = x − x at the
2
(a) yt = x + at 2 (b) y = xt + at 2
point where x = 1 , is [MP PET 1992]
(a) y = − x − 1 (b) y = − x + 1 a
(c) y = xt + (d) y = tx
t
(c) y = x + 1 (d) y = x − 1
73. The point of intersection of the latus rectum and axis of the 83. The line y = 2 x + c is a tangent to the parabola y 2 = 16 x , if
parabola y 2 + 4 x + 2 y − 8 = 0 c equals [MNR 1988]
(a) −2 (b) −1
(a) (5/4, –1) (b) (9/4, –1)
(c) (7/2, 5/2) (d) None of these (c) 0 (d) 2
74. The point of contact of the tangent 18 x − 6 y + 1 = 0 to the 84. The line y = mx + 1 is a tangent to the parabola y 2 = 4 x , if
[MNR 1990; Kurukshetra CEE 1998;
parabola y 2 = 2 x is
DCE 2000; Pb. CET 2004]
 −1 −1   −1 1 
(a)  ,  (b)  , 
 18 3   18 3  (a) m = 1 (b) m = 2
 1 −1   1 1 (c) m = 4 (d) m = 3
(c)  ,  (d)  , 
 18 3   18 3  85. The angle of intersection between the curves y 2 = 4 x and
75. The equation of the common tangent of the parabolas x 2 = 32 y at point (16, 8), is [RPET 1987, 96]
x 2 = 108 y and y 2 = 32 x , is
3 4
(a) 2 x + 3 y = 36 (b) 2 x + 3 y + 36 = 0 (a) tan −1   (b) tan −1  
5 5
(c) 3 x + 2 y = 36 (d) 3 x + 2 y + 36 = 0

(c)  (d)
76. The line lx + my + n = 0 will touch the parabola y 2 = 4 ax , 2
if [RPET 1988; MNR 1977; MP PET 2003] 86. The locus of a foot of perpendicular drawn to the tangent of
(a) mn = al 2 (b) lm = an 2 parabola y 2 = 4 ax from focus, is [RPET 1989]
(c) ln = am 2 (d) mn = al (a) x = 0 (b) y = 0
77. The line x cos  + y sin  = p will touch the parabola
(c) y = 2 a(x + a)
2
(d) x 2 + y 2 (x + a) = 0
y 2 = 4 a(x + a) , if
87. If the straight line x +y =1 touches the parabola
(a) p cos  + a = 0 (b) p cos  − a = 0
y − y + x = 0 , then the co-ordinates of the point of contact
2
(c) a cos  + p = 0 (d) a cos  − p = 0
are [RPET 1991]
78. The equation of a tangent to the parabola y 2 = 4 ax making 1 1
an angle  with x-axis is (a) (1, 1) (b)  , 
2 2
(a) y = x cot  + a tan  (b) x = y tan  + a cot 
(c) (0, 1) (d) (1, 0)
(c) y = x tan  + a cot  (d) None of these 88. If the line y = mx + c is a tangent to the parabola
79. The equation of the tangent to the parabola y 2 = 4 x + 5 a
y 2 = 4 a(x + a) then ma + is equal to
parallel to the line y = 2 x + 7 is [MNR 1979] m
(a) 2 x − y − 3 = 0 (b) 2 x − y + 3 = 0 (a) c (b) 2c
(c) 2 x + y + 3 = 0 (d) None of these (c) – c (d) 3c
 Conic Sections 733

89. A tangent to the parabola y 2 = 8 x makes an angle of 45 o 98. The condition for which the straight line y = mx + c touches
with the straight line y = 3 x + 5 , then the equation of the parabola y 2 = 4 ax is [MP PET 1997, 2001]
tangent is a
(a) a = c (b) =m
(a) 2 x + y − 1 = 0 (b) x + 2 y − 1 = 0 c
(c) 2 x + y + 1 = 0 (d) None of these (c) m = a 2 c (d) m = ac 2
90. The angle between the tangents drawn at the end points of 99. If the parabola y 2 = 4 ax passes through the point (1, –2),
the latus rectum of parabola y 2 = 4 ax , is then the tangent at this point is [MP PET 1998]
 2 (a) x + y − 1 = 0 (b) x − y − 1 = 0
(a) (b)
3 3 (c) x + y + 1 = 0 (d) x − y + 1 = 0
  100. The equation of the tangent to the parabola y 2 = 16 x , which
(c) (d)
4 2 is perpendicular to the line y = 3 x + 7 is
91. The line y = mx + c touches the parabola x 2 = 4 ay , if [MP PET 1998]
[MNR 1973; MP PET 1994, 99] (a) y − 3 x + 4 = 0 (b) 3 y − x + 36 = 0
(a) c = −am (b) c = −a / m (c) 3 y + x − 36 = 0 (d) 3 y + x + 36 = 0

(c) c = −am 2
(d) c = a / m 2 101. The equation of the tangent to the parabola y 2 = 4 ax at
92. The locus of the point of intersection of the perpendicular point (a / t 2 , 2 a / t) is [RPET 1996]
tangents to the parabola x 2 = 4 ay is [MP PET 1994] (a) ty = xt 2 + a (b) ty = x + at 2
(a) Axis of the parabola
(c) y = tx + at 2 (d) y = tx + (a / t 2 )
(b) Directrix of the parabola
(c) Focal chord of the parabola 102. The equation of common tangent to the circle x 2 + y 2 = 2
(d) Tangent at vertex to the parabola and parabola y 2 = 8 x is [RPET 1997]
93. The angle between the tangents drawn from the origin to the (a) y = x + 1 (b) y = x + 2
parabola y 2 = 4 a(x − a) is [MNR 1994] (c) y = x − 2 (d) y = − x + 2
(a) 90 o
(b) 30 o 103. If the line lx + my + n = 0 is a tangent to the parabola
1 y 2 = 4 ax , then locus of its point of contact is [RPET 1997]
(c) tan −1 (d) 45 o
2 (a) A straight line (b) A circle
94. If line x = my + k touches the parabola x 2 = 4 ay , then k = (c) A parabola (d) Two straight lines
[MP PET 1995] 104. The line x − y + 2 = 0 touches the parabola y 2 = 8 x at the
a point [Roorkee 1998]
(a) (b) am
m (a) (2, − 4 ) (b) (1, 2 2 )
(c) am 2
(d) − am 2
(c) (4 , − 4 2 ) (d) (2, 4)
95. If y 1 , y 2 are the ordinates of two points P and Q on the
105. The tangent to the parabola y 2 = 4 ax at the point (a, 2a)
parabola and y 3 is the ordinate of the point of intersection
makes with x-axis an angle equal to [SCRA 1996]
of tangents at P and Q, then
 
(a) y 1 , y 2 , y 3 are in A.P. (b) y 1 , y 3 , y 2 are in A.P. (a) (b)
3 4
(c) y 1 , y 2 , y 3 are in G.P. (d) y 1 , y 3 , y 2 are in G.P.  
(c) (d)
96. The two parabolas y 2 = 4 x and x 2 = 4 y intersect at a 2 6
point P, whose abscissa is not zero, such that 106. If lx + my + n = 0 is tangent to the parabola x 2 = y , then
(a) They both touch each other at P condition of tangency is [RPET 1999]
(b) They cut at right angles at P (a) l 2 = 2mn (b) l = 4 m 2 n 2
(c) The tangents to each curve at P make complementary (c) m 2 = 4 ln (d) l 2 = 4 mn
angles with the x-axis
107. The equation of the tangent to the parabola y 2 = 9 x which
(d) None of these
goes through the point (4, 10), is [MP PET 2000]
97. The line y = 2 x + c is tangent to the parabola y 2 = 4 x ,
(a) x + 4 y + 1 = 0 (b) 9 x + 4 y + 4 = 0
then c = [MP PET 1996]
(c) x − 4 y + 36 = 0 (d) 9 x − 4 y + 4 = 0
1 1
(a) − (b) 108. Two perpendicular tangents to y 2 = 4 ax always intersect
2 2
on the line, if [Karnataka CET 2000]
1
(c) (d) 4 (a) x = a (b) x + a = 0
3
(c) x + 2 a = 0 (d) x + 4 a = 0
 734 Conic Sections
109. The equation of the common tangent touching the circle 119. The locus of the middle points of the chords of the parabola
(x − 3) 2 + y 2 = 9 and the parabola y 2 = 4 x above the x- y 2 = 4 ax which passes through the origin
axis, is [IIT Screening 2001] [RPET 1997; UPSEAT 1999]
(a) 3y = 3x + 1 (b) 3 y = −(x + 3) (a) y = ax2
(b) y 2 = 2 ax
(c) 3y = x + 3 (d) 3 y = −(3 x + 1) (c) y 2 = 4 ax (d) x 2 = 4 ay
110. The point at which the line y = mx + c touches the parabola 120. The point on the parabola y 2 = 8 x at which the normal is
y = 4 ax is
2
[RPET 2001] parallel to the line x − 2 y + 5 = 0 is
 a 2a   a −2 a  (a) (−1 / 2, 2) (b) (1 / 2, − 2)
(a)  2 ,  (b)  2 , 
m m  m m 
(c) (2, − 1 / 2) (d) (−2, 1 / 2)
 a 2a   a 2a 
121. The maximum number of normal that can be drawn from a
(c)  − 2 ,  (d)  − 2 , − 
 m m   m m  point to a parabola is [MP PET 1990]
111. The tangent drawn at any point P to the parabola y 2 = 4 ax (a) 0 (b) 1
meets the directrix at the point K, then the angle which KP (c) 2 (d) 3
subtends at its focus is [RPET 1996, 2002] 122. The point on the parabola y 2 = 8 x at which the normal is
(a) 30o (b) 45o inclined at 60o to the x-axis has the co-ordinates
(c) 60o (d) 90o [MP PET 1993]
112. The point of intersection of the parabola at the points t 1 and
(a) (6, − 4 3 ) (b) (6, 4 3 )
t 2 is [RPET 2002]
(a) (at 1 t 2 , a(t 1 + t 2 )) (b) (2 at 1 t 2 , a(t 1 + t 2 )) (c) (−6, − 4 3 ) (d) (−6, 4 3 )
(c) (2 at 1 t 2 , 2 a(t 1 + t 2 )) (d) None of these 123. The slope of the normal at the point (at 2 , 2 at) of the
113. The angle of intersection between the curves x 2 = 4 (y + 1) parabola y 2 = 4 ax , is [MNR 1991; UPSEAT 2000]
and x 2
= −4 (y + 1) is [UPSEAT 2002] 1
(a) (b) t
  t
(a) (b)
6 4 1
(c) –t (d) −
 t
(c) 0 (d)
2 a 
124. The equation of the normal at the point  , a  to the
114. Angle between two curves y = 4 (x + 1) and x 2 = 4 (y + 1)
2
4 
is [UPSEAT 2002] parabola y 2 = 4 ax , is [RPET 1984]
(a) 0o (b) 90o
(a) 4 x + 8 y + 9 a = 0 (b) 4 x + 8 y − 9 a = 0
(c) 60o (d) 30o
(c) 4 x + y − a = 0 (d) 4 x − y + a = 0
115. If The tangent to the parabola y = ax makes an angle of 45o
2

with x-axis, then the point of contact is 125. The equation of normal to the parabola at the point
[RPET 1985, 90, 2003]  a 2a 
 2,  ,is [RPET 1987]
a a a a m m 
(a)  ,  (b)  , 
2 2 4 4 (a) y = m 2 x − 2mx − am 3 (b) m 3 y = m 2 x − 2 am 2 − a
a a a a (c) m 3 y = 2 am 2 − m 2 x + a (d) None of these
(c)  ,  (d)  , 
2 4 4 2
126. If the line 2 x + y + k = 0 is normal to the parabola
116. Tangents at the extremities of any focal chord of a parabola
intersect y 2 = −8 x , then the value of k will be [RPET 1986, 97]
(a) At right angles (b) On the directrix (a) −16 (b) −8
(c) On the tangents at vertex (d) None of these (c) −24 (d) 24
117. The point of intersection of tangents at the ends of the latus-
127. If a normal drawn to the parabola y 2 = 4 ax at the point
rectum of the parabola y 2 = 4 x is equal to [Pb. CET 2003]
(a, 2 a) meets parabola again on (at 2 , 2 at) , then the value of
(a) (1, 0) (b) (–1, 0)
(c) (0, 1) (d) (0, –1) t will be [RPET 1990]
118. The angle between the tangents drawn from the points (1,4) (a) 1 (b) 3
to the parabola y 2 = 4 x is [IIT Screening 2004] (c) –1 (d) –3
  128. In the parabola y 2 = 6 x , the equation of the chord through
(a) (b) vertex and negative end of latus rectum, is
2 3
  (a) y = 2 x (b) y + 2 x = 0
(c) (d)
4 6 (c) x = 2 y (d) x + 2 y = 0
 Conic Sections 735

129. The length of chord of contact of the tangents drawn from the 139. Tangents drawn at the ends of any focal chord of a parabola
point (2, 5) to the parabola y 2 = 8 x , is [MNR 1976] y 2 = 4 ax intersect in the line
1 (a) y − a = 0 (b) y + a = 0
(a) 41 (b) 41
2 (c) x − a = 0 (d) x + a = 0
3 140. The centroid of the triangle formed by joining the feet of the
(c) 41 (d) 2 41
2 normals drawn from any point to the parabola y 2 = 4 ax ,
130. If ‘a’ and ‘c’ are the segments of a focal chord of a parabola lies on [MP PET 1999]
and b the semi-latus rectum, then [MP PET 1995] (a) Axis (b) Directrix
(a) a, b, c are in A.P. (b) a, b, c are in G.P. (c) Latus rectum (d) Tangent at vertex
(c) a, b, c are in H.P. (d) None of these 141. If the normal to y 2 = 12 x at (3, 6) meets the parabola again
131. If the segment intercepted by the parabola y = 4 ax with
2 in (27, –18) and the circle on the normal chord as diameter
the line lx + my + n = 0 subtends a right angle at the vertex, is [Kurukshetra CEE 1998]

then (a) x 2 + y 2 + 30 x + 12 y − 27 = 0
(a) 4 al + n = 0 (b) 4 al + 4 am + n = 0 (b) x 2 + y 2 + 30 x + 12 y + 27 = 0
(c) 4 am + n = 0 (d) al + n = 0 (c) x 2 + y 2 − 30 x − 12 y − 27 = 0
132. A set of parallel chords of the parabola y 2 = 4 ax have their
(d) x 2 + y 2 − 30 x + 12 y − 27 = 0
mid-point on
(a) Any straight line through the vertex 142. The length of the normal chord to the parabola y 2 = 4 x ,
(b) Any straight line through the focus which subtends right angle at the vertex is [RPET 1999]
(c) Any straight line parallel to the axis (a) 6 3 (b) 3 3
(d) Another parabola (c) 2 (d) 1
133. The equations of the normals at the ends of latus rectum of 143. If x + y = k is a normal to the parabola y 2 = 12 x , then k is
the parabola y 2 = 4 ax are given by [IIT Screening 2000]
(a) x − y − 6 ax + 9 a = 0
2 2 2 (a) 3 (b) 9
(c) –9 (d) –3
(b) x 2 − y 2 − 6 ax − 6 ay + 9 a 2 = 0
144. The normal at the point (bt 12 , 2bt 1 ) on a parabola meets the
(c) x − y − 6 ay + 9 a = 0
2 2 2
parabola again in the point (bt 22 , 2bt 2 ) , then
(d) None of these
[MNR 1986; RPET 2003; AIEEE 2003]
134. If the normals at two points P and Q of a parabola y 2 = 4 ax 2 2
(a) t 2 = −t 1 − (b) t 2 = −t 1 +
intersect at a third point R on the curve, then the product of t1 t1
ordinates of P and Q is
2 2
(a) 4a 2 (b) 2a 2 (c) t 2 = t 1 − (d) t 2 = t 1 +
t1 t1
(c) − 4a 2 (d) 8a 2
145. The focal chord to y 2 = 16 x is tangent to (x − 6 ) 2 + y 2 = 2 ,
135. If x = my + c is a normal to the parabola x 2 = 4 ay , then the
then the possible value of the slope of this chord, are [IIT
value of c is Screening 2003]
(a) − 2 am − am 3 (b) 2 am + am 3 (a) {−1, 1} (b) {–2, 2}
(c) −
2a
−
a
(d)
2a
+
a (c) {-2, 1/2} (d) {2, –1/2}
m m3 m m3 146. The normal to the parabola y = 8 x at the point (2, 4) meets
2

136. If PSQ is the focal chord of the parabola y 2 = 8 x such that the parabola again at the point [Orissa JEE 2003]
SP = 6 . Then the length SQ is (a) {–18, –12} (b) {–18, 12}
(a) 6 (b) 4 (c) {18, 12} (d) (18, –12)
(c) 3 (d) None of these 147. The polar of focus of parabola [RPET 1999]
(a) x-axis (b) y-axis
137. At what point on the parabola y 2 = 4 x , the normal makes
(c) Directrix (d) Latus rectum
equal angles with the co-ordinate axes [RPET 1994]
148. Equation of diameter of parabola y 2 = x corresponding to
(a) (4, 4) (b) (9, 6)
(c) (4, –4) (d) (1, –2) the chord x − y + 1 = 0 is [RPET 2003]

138. Equation of any normal to the parabola y 2 = 4 a(x − a) is (a) 2 y = 3 (b) 2 y = 1

(c) 2 y = 5 (d) y = 1
(a) y = mx − 2 am − am 3
149. The area of the triangle formed by the lines joining the vertex
(b) y = m (x + a) − 2 am − am 3 of the parabola x 2 = 12 y to the ends of its latus rectum is
(c) y = m (x − a) +
a (a) 12 sq. unit (b) 16 sq. unit
m (c) 18 sq. unit (d) 24 sq. unit
(d) y = m (x − a) − 2 am − am 3
 736 Conic Sections
150. The area of triangle formed inside the parabola y 2 = 4 x and 159. The angle of intersection between the curves x 2 = 8 y and
whose ordinates of vertices are 1, 2 and 4 will be y 2 = 8 x at origin is [RPET 1997]
[RPET 1990]
(a) /4 (b) /3
(a)
7
(b)
5 (c) /6 (d) /2
2 2
160. If the line y = 2 x + k is a tangent to the curve x 2 = 4 y , then
3 3 k is equal to [AMU 2002]
(c) (d)
2 4 (a) 4 (b) 1/2
151. An equilateral triangle is inscribed in the parabola y 2 = 4 ax (c) –4 (d) –1/2
whose vertices are at the parabola, then the length of its side 161. The equation to a parabola which passes through the
is equal to intersection of a straight line x + y = 0 and the circle
x 2 + y 2 + 4 y = 0 is [Orissa JEE 2005]
(a) 8a (b) 8a 3
(a) y = 4 x
2
(b) y = x
2
(c) a 2 (d) None of these
(c) y = 2 x
2
(d) None of these
152. The ordinates of the triangle inscribed in parabola y 2 = 4 ax
are y 1 , y 2 , y 3 , then the area of triangle is 162. Let a circle tangent to the directrix of a parabola y 2 = 2 ax
has its centre coinciding with the focus of the parabola. Then
1 the point of intersection of the parabola and circle is
(a) (y 1 + y 2 )(y 2 + y 3 )(y 3 + y 1 )
8a [Orissa JEE 2005]
1 (a) (a, –a) (b) (a / 2, a / 2)
(b) (y 1 + y 2 )(y 2 + y 3 )(y 3 + y 1 )
4a (c) (a / 2,  a) (d) ( a, a / 2)

(c)
1
(y 1 − y 2 )(y 2 − y 3 )(y 3 − y 1 ) 163. The length intercepted by the curve y 2 = 4 x on the line
8a satisfying dy / dx = 1 and passing through point (0, 1) is
1 given by [Orissa JEE 2005]
(d) (y 1 − y 2 )(y 2 − y 3 )(y 3 − y 1 )
4a (a) 1 (b) 2
153. From the point (–1, 2) tangent lines are drawn to the (c) 0 (d) None of these
parabola y 2 = 4 x , then the equation of chord of contact is 164. The equation of a straight line drawn through the focus of the
parabola y 2 = −4 x at an angle of 120° to the x-axis is
[Roorkee 1994]
[Orissa JEE 2005]
(a) y = x + 1 (b) y = x − 1
(a) y + 3 (x − 1) = 0 (b) y − 3 (x − 1) = 0
(c) y + x = 1 (d) None of these
154. For the above problem, the area of triangle formed by chord (c) y + 3 (x + 1) = 0 (d) y − 3 (x + 1) = 0
of contact and the tangents is given by [Roorkee 1994] 165. The number of parabolas that can be drawn if two ends of the
latus rectum are given [DCE 2005]
(a) 8 (b) 8 3
(a) 1 (b) 2
(c) 8 2 (d) None of these (c) 4 (d) 3
155. The point on parabola 2 y = x , which is nearest to the point
2 166. The normal meet the parabola y 2 = 4 ax at that point where
(0, 3) is [J & K 2005] the abissiae of the point is equal to the ordinate of the point
is [DCE 2005]
(a) (4, 8) (b) (1, 1 / 2)
(a) (6 a, − 9 a) (b) (−9 a, 6 a)
(c) (2, 2) (d) None of these
(c) (−6 a, 9 a) (d) (9 a, − 6 a)
156. From the point (–1, –60) two tangents are drawn to the
parabola y 2 = 4 x . Then the angle between the two tangents Ellipse
is [J & K 2005]
1. If the latus rectum of an ellipse be equal to half of its minor
(a) 30° (b) 45° axis, then its eccentricity is
(c) 60° (d) 90° [MP PET 1991, 97; Karnataka CET 2000]
157. The ends of the latus rectum of the conic (a) 3/2 (b) 3 /2
x 2 + 10 x − 16 y + 25 = 0 are [Karnataka CET 2005]
(c) 2/3 (d) 2 / 3
(a) (3, –4), (13, 4) (b) (–3, –4), (13, –4)
2. If distance between the directrices be thrice the distance
(c) (3, 4), (–13, 4) (d) (5, –8), (–5, 8) between the foci, then eccentricity of ellipse is
158. Tangent to the parabola y = x 2 + 6 at (1, 7) touches the (a) 1/2 (b) 2/3
circle x 2 + y 2 + 16 x + 12 y + c = 0 at the point (c) 1 / 3 (d) 4/5
[IIT Screening 2005] 3. The equation of the ellipse whose centre is at origin and
(a) (–6, –9) (b) (–13, –9) which passes through the points (–3, 1) and (2, –2) is
(c) (–6, –7) (d) (13, 7) (a) 5 x 2 + 3 y 2 = 32 (b) 3 x 2 + 5 y 2 = 32
 Conic Sections 737

(c) 5 x 2 − 3 y 2 = 32 (d) 3 x 2 + 5 y 2 + 32 = 0 (c) 5/3 (d) 10/3

4. If the eccentricity of an ellipse be 5/8 and the distance 15. If the distance between a focus and corresponding directrix
between its foci be 10, then its latus rectum is of an ellipse be 8 and the eccentricity be 1/2, then length of
the minor axis is
(a) 39/4 (b) 12
(c) 15 (d) 37/2 (a) 3 (b) 4 2
5. If the foci and vertices of an ellipse be (1, 0 ) and (2, 0 ) , (c) 6 (d) None of these
then the minor axis of the ellipse is 16. Eccentricity of the conic 16 x + 7 y 2 = 112 is
2
[MNR 1981]
(a) 2 5 (b) 2
(a) 3 / 7 (b) 7/16
(c) 4 (d) 2 3
(c) 3/4 (d) 4/3
6. The equations of the directrices of the ellipse
17. If the distance between the foci of an ellipse be equal to its
16 x 2 + 25 y 2 = 400 are minor axis, then its eccentricity is
(a) 2 x = 25 (b) 5 x = 9
(a) 1/2 (b) 1 / 2
(c) 3 x = 10 (d) None of these
7. The eccentricity of an ellipse is 2/3, latus rectum is 5 and (c) 1/3 (d) 1 / 3
centre is (0, 0). The equation of the ellipse is 18. An ellipse passes through the point (–3, 1) and its
x2 y2 4 x 2 4y 2
(a) + =1 (b) + =1 eccentricity is
2
. The equation of the ellipse is
81 45 81 45 5
x2 y2 x2 y2
(c) + =1 (d) + =5 (a) 3 x 2 + 5 y 2 = 32 (b) 3 x 2 + 5 y 2 = 25
9 5 81 45
8. The latus rectum of an ellipse is 10 and the minor axis is (c) 3 x 2 + y 2 = 4 (d) 3 x 2 + y 2 = 9
equal to the distance between the foci. The equation of the 19. The lengths of major and minor axis of an ellipse are 10 and
ellipse is 8 respectively and its major axis along the y-axis. The
(a) x 2 + 2 y 2 = 100 (b) x 2 + 2 y 2 = 10 equation of the ellipse referred to its centre as origin is
[Pb. CET 2003]
(c) x 2 − 2 y 2 = 100 (d) None of these
2 2 2 2
9. The distance between the directrices of the ellipse x y x y
(a) + =1 (b) + =1
25 16 16 25
x2 y2
+ = 1 is
36 20 x2 y2 x2 y2
(c) + =1 (d) + =1
(a) 8 (b) 12 100 64 64 100
(c) 18 (d) 24 20. If the centre, one of the foci and semi-major axis of an ellipse
10. The distance between the foci of the ellipse 3 x + 4 y = 48
2 2 be (0, 0), (0, 3) and 5 then its equation is
is [AMU 1981]
(a) 2 (b) 4 x 2
y 2
x2
y 2
(a) + =1 (b) + =1
(c) 6 (d) 8 16 25 25 16
11. The equation of the ellipse whose vertices are (5, 0 ) and
x2 y2
foci are (4 , 0 ) is (c) + =1 (d) None of these
9 25
(a) 9 x 2 + 25 y 2 = 225 (b) 25 x 2 + 9 y 2 = 225 21. The equation of the ellipse whose one of the vertices is (0,7)
(c) 3 x + 4 y = 192
2 2
(d) None of these and the corresponding directrix is y = 12 , is
12. The equation of the ellipse whose foci are (5, 0 ) and one of (a) 95 x 2 + 144 y 2 = 4655 (b) 144 x 2 + 95 y 2 = 4655
its directrix is 5 x = 36 , is (c) 95 x 2 + 144 y 2 = 13680 (d) None of these
2 2 2 2
x y x y
(a) + =1 (b) + =1 22. The equation 2 x 2 + 3 y 2 = 30 represents [MP PET 1988]
36 11 6 11
(a) A circle (b) An ellipse
x2 y2
(c) + =1 (d) None of these (c) A hyperbola (d) A parabola
6 11
23. The equation of the ellipse whose latus rectum is 8 and
13. If the eccentricity of an ellipse be 1 / 2 , then its latus 1
rectum is equal to its whose eccentricity is , referred to the principal axes of
2
(a) Minor axis (b) Semi-minor axis coordinates, is [MP PET 1993]
(c) Major axis (d) Semi-major axis
x2 y2 x2 y2
14. The length of the latus rectum of the ellipse 5 x 2 + 9 y 2 = 45 (a) + =1 (b) + =1
18 32 8 9
is [MNR 1978, 80, 81]
(a) 5 /4 (b) 5 /2
 738 Conic Sections
x2 y2 x2 y2 (a) x 2 + y 2 = a 2 − b 2 (b) x 2 − y 2 = a 2 − b 2
(c) + =1 (d) + =1
64 32 16 24 (c) x 2 + y 2 = a 2 + b 2 (d) x 2 − y 2 = a 2 + b 2
24. Eccentricity of the ellipse whose latus rectum is equal to the
distance between two focus points, is x2 y2
32. The length of the latus rectum of the ellipse + =1
36 49
5 +1 5 −1
(a) (b) [Karnataka CET 1993]
2 2
(a) 98/6 (b) 72/7
5 3 (c) 72/14 (d) 98/12
(c) (d)
2 2 x2 y2
33. The distance of the point '  ' on the ellipse + =1
25. For the ellipse 3 x 2 + 4 y 2 = 12 , the length of latus rectum is a2 b2
[MNR 1973] from a focus is
3 (a) a(e + cos  ) (b) a(e − cos  )
(a) (b) 3
2 (c) a(1 + e cos  ) (d) a(1 + 2 e cos  )
8 3 34. The equation of the ellipse whose one focus is at (4, 0) and
(c) (d) whose eccentricity is 4/5, is [Karnataka CET 1993]
3 2
x2 y2 x2 y2
x2 y2 (a) + =1 (b) + =1
26. For the ellipse + = 1 , the eccentricity is [MNR 1974] 32 52 52 32
64 28
x2 y2 x2 y2
3 4 (c) + =1 (d) + =1
(a) (b) 5 2
4 2
4 2
52
4 3
2 1 35. The foci of 16 x 2 + 25 y 2 = 400 are [BIT Ranchi 1996]
(c) (d)
7 3 (a) (3, 0 ) (b) (0,  3)
27. If the length of the major axis of an ellipse is three times the (c) (3, − 3) (d) (−3, 3)
length of its minor axis, then its eccentricity is
36. Eccentricity of the ellipse 9 x + 25 y 2 = 225 is
2
[EAMCET 1990]
[Kerala (Engg.) 2002]
1 1
(a) (b) 3 4
3 3 (a) (b)
5 5
1 2 2
(c) (d) 9 34
2 3 (c) (d)
25 5
1 37. The eccentricity of the ellipse 25 x 2 + 16 y 2 = 100 , is
28. The length of the latus rectum of an ellipse is of the major
3
5 4
axis. Its eccentricity is [EAMCET 1991] (a) (b)
14 5
2 2
(a) (b) 3 2
3 3 (c) (d)
5 5
4
543 3 38. The length of the latus rectum of the ellipse 9 x 2 + 4 y 2 = 1 ,
(c) (d)  
7 3
4 is [MP PET 1999]
29. An ellipse is described by using an endless string which is 3 8
(a) (b)
passed over two pins. If the axes are 6 cm and 4 cm, the 2 3
necessary length of the string and the distance between the
4 8
pins respectively in cm, are [MNR 1989] (c) (d)
9 9
(a) 6, 2 5 (b) 6, 5
39. The locus of a variable point whose distance from (–2, 0) is
(c) 4 , 2 5 (d) None of these 2 9
times its distance from the line x = − , is [IIT 1994]
3 2
x2 y2
30. The equation + + 1 = 0 represents an ellipse, if (a) Ellipse (b) Parabola
2−r r−5 (c) Hyperbola (d) None of these
[MP PET 1995]
40. If P  (x , y ) , F1  (3, 0 ) , F2  (−3, 0 ) and
(a) r  2 (b) 2  r  5
16 x 2 + 25 y 2 = 400 , then PF1 + PF2 equals [IIT 1998]
(c) r  5 (d) None of these
31. The locus of the point of intersection of perpendicular (a) 8 (b) 6
x2 y2 (c) 10 (d) 12
tangents to the ellipse 2 + = 1 , is [MP PET 1995]
a b2
 Conic Sections 739

41. P is any point on the ellipse 9 x 2 + 36 y 2 = 324 , whose foci (a) 2a (b)
2a
are S and S’. Then SP + S ' P equals [DCE 1999] b
(a) 3 (b) 12 2b b2
(c) (d)
(c) 36 (d) 324 a a
42. What is the equation of the ellipse with foci (2, 0 ) and 51. The equation of ellipse whose distance between the foci is
1 equal to 8 and distance between the directrix is 18, is
eccentricity = [DCE 1999]
2 (a) 5 x 2 − 9 y 2 = 180 (b) 9 x 2 + 5 y 2 = 180
(a) 3 x + 4 y = 48
2 2
(b) 4 x + 3 y = 48
2 2
(c) x 2 + 9 y 2 = 180 (d) 5 x 2 + 9 y 2 = 180
(c) 3 x 2 + 4 y 2 = 0 (d) 4 x 2 + 3 y 2 = 0 52. In an ellipse the distance between its foci is 6 and its minor
axis is 8. Then its eccentricity is [EAMCET 1994]
43. The eccentricity of the ellipse 4 x 2 + 9 y 2 = 36 , is
4 1
[MP PET 2000] (a) (b)
5 52
1 1
(a) (b)
3 1
2 3 3 (c) (d)
5 2
5 5
(c) (d) 53. If a bar of given length moves with its extremities on two
3 6 fixed straight lines at right angles, then the locus of any point
44. The eccentricity of the ellipse 25 x 2 + 16 y 2 = 400 is on bar marked on the bar describes a/an [Orissa JEE 2003]
[MP PET 2001] (a) Circle (b) Parabola
(a) 3/5 (b) 1/3 (c) Ellipse (d) Hyperbola
(c) 2/5 (d) 1/5 54. The centre of the ellipse 4 x 2 + 9 y 2 − 16 x − 54 y + 61 = 0 is
45. The distance between the foci of an ellipse is 16 and [MP PET 1992]
1 (a) (1,3) (b) (2, 3)
eccentricity is . Length of the major axis of the ellipse is
2 (c) (3, 2) (d) (3, 1)
[Karnataka CET 2001]
55. Latus rectum of ellipse 4 x 2 + 9 y 2 − 8 x − 36 y + 4 = 0 is
(a) 8 (b) 64
[MP PET 1989]
(c) 16 (d) 32
(a) 8/3 (b) 4/3
x2 y2
46. If the eccentricity of the two ellipse + = 1 and 5
169 25 (c) (d) 16/3
3
x2 y2
+ = 1 are equal, then the value of a / b is
a2 b2 56. Eccentricity of the ellipse 4 x 2 + y 2 − 8 x + 2 y + 1 = 0 is
[UPSEAT 2001] (a) 1 / 3 (b) 3 /2
(a) 5/13 (b) 6/13
(c) 1 / 2 (d) None of these
(c) 13/5 (d) 13/6
57. The equation of an ellipse whose eccentricity is 1/2 and the
47. In the ellipse, minor axis is 8 and eccentricity is
5
. Then vertices are (4, 0) and (10, 0) is
3 (a) 3 x 2 + 4 y 2 − 42 x + 120 = 0
major axis is [Karnataka CET 2002]
(a) 6 (b) 12 (b) 3 x 2 + 4 y 2 + 42 x + 120 = 0
(c) 10 (d) 16 (c) 3 x 2 + 4 y 2 + 42 x − 120 = 0
48. In an ellipse 9 x 2 + 5 y 2 = 45 , the distance between the foci (d) 3 x 2 + 4 y 2 − 42 x − 120 = 0
is [Karnataka CET 2002]
58. The equation of the ellipse whose centre is (2, –3), one of the
(a) 4 5 (b) 3 5 foci is (3, –3) and the corresponding vertex is (4, –3) is
(c) 3 (d) 4 (x − 2) 2 (y + 3) 2 (x − 2) 2 (y + 3) 2
1 (a) + =1 (b) + =1
49. Equation of the ellipse with eccentricity and foci at (1, 0 ) 3 4 4 3
2
x2 y2
is [MP PET 2002] (c) + =1 (d) None of these
2 2 2 2 3 4
x y x y
(a) + =1 (b) + =1 59. The equation 14 x 2 − 4 xy + 11 y 2 − 44 x − 58 y + 71 = 0
3 4 4 3
represents [BIT Ranchi 1986]
x2 y2 4
(c) + = (d) None of these (a) A circle (b) An ellipse
3 4 3
(c) A hyperbola (d) A rectangular hyperbola
50. The sum of focal distances of any point on the ellipse with
major and minor axes as 2a and 2b respectively, is equal to (x + y − 2) 2 (x − y ) 2
60. The centre of the ellipse + = 1 is
[MP PET 2003] 9 16
[EAMCET 1994]
 740 Conic Sections
(a) (0, 0) (b) (1, 1) (a) 4/5 (b) 3/5
(c) (1, 0) (d) (0, 1) (c) 5/4 (d) Imaginary
61. The equation of an ellipse whose focus (–1, 1), whose 71. The length of the axes of the conic
directrix is x − y + 3 = 0 and whose eccentricity is
1
, is 9 x 2 + 4 y 2 − 6 x + 4 y + 1 = 0 , are [Orissa JEE 2002]
2
1 2
given by [MP PET 1993] (a) ,9 (b) 3,
2 5
(a) 7 x 2 + 2 xy + 7 y 2 + 10 x − 10 y + 7 = 0
2
(c) 1, (d) 3, 2
(b) 7 x 2 − 2 xy + 7 y 2 − 10 x + 10 y + 7 = 0 3
(c) 7 x 2 − 2 xy + 7 y 2 − 10 x − 10 y − 7 = 0 72. The eccentricity of the ellipse
9 x 2 + 5 y 2 − 18 x − 2 y − 16 = 0 is [EAMCET 2003]
(d) 7 x 2 − 2 xy + 7 y 2 + 10 x + 10 y − 7 = 0
(a) 1/2 (b) 2/3
62. The foci of the ellipse 25 (x + 1) 2 + 9(y + 2) 2 = 225 are at (c) 1/3 (d) 3/4
[MNR 1991; MP PET 1998; UPSEAT 2000]
73. The eccentricity of the conic 4 x 2 + 16 y 2 − 24 x − 3 y = 1 is
(a) (–1, 2) and (–1, –6) (b) (–1, 2) and (6, 1)
[MP PET 2004]
(c) (1, –2) and (1, –6) (d) (–1, –2) and (1, 6)
63. The eccentricity of the ellipse 9 x 2 + 5 y 2 − 30 y = 0 , is 3 1
(a) (b)
2 2
[MNR 1993; Pb. CET 2004]
(a) 1/3 (b) 2/3 3
(c) (d) 3
(c) 3/4 (d) None of these 4
64. The curve represented by x = 3(cos t + sin t) , 74. If the line y = 2x + c be a tangent to the ellipse
y = 4 (cos t − sin t) is [EAMCET 1988; DCE 2000] x 2
y 2
+ = 1 , then c = [MNR 1979; DCE 2000]
(a) Ellipse (b) Parabola 8 4
(c) Hyperbola (d) Circle (a) 4 (b) 6
65. Equation x = a cos  , y = b sin  (a  b ) represent a conic (c) 1 (d) 8
section whose eccentricity e is given by 75. The position of the point (4, –3) with respect to the ellipse
a +b
2 2
a +b
2 2
2 x 2 + 5 y 2 = 20 is
(a) e 2 = (b) e 2 =
a2 b2 (a) Outside the ellipse (b) On the ellipse
a −b
2 2
a −b
2 2
(c) On the major axis (d) None of these
(c) e 2 = (d) e 2 =
a2 b2 76. The equation of the tangent to the ellipse x 2 + 16 y 2 = 16
66. The eccentricity of the ellipse making an angle of 60 o with x-axis is
4 x + 9 y + 8 x + 36 y + 4 = 0 is
2 2
[MP PET 1996]
(a) 3x − y + 7 = 0 (b) 3x − y − 7 = 0
5 3
(a) (b) (c) 3x − y  7 = 0 (d) None of these
6 5
77. The position of the point (1, 3) with respect to the ellipse
2 5
(c) (d) 4 x 2 + 9 y 2 − 16 x − 54 y + 61 = 0 [MP PET 1991]
3 3
67. The co-ordinates of the foci of the ellipse (a) Outside the ellipse (b) On the ellipse
3 x 2 + 4 y 2 − 12 x − 8 y + 4 = 0 are (c) On the major axis (d) On the minor axis
78. The line lx + my − n = 0 will be tangent to the ellipse
(a) (1, 2), (3, 4) (b) (1, 4), (3, 1)
(c) (1, 1), (3, 1) (d) (2, 3), (5, 4) x2 y2
2
+ = 1 , if
68. The eccentricity of the curve represented by the equation a b2
x 2 + 2 y 2 − 2 x + 3 y + 2 = 0 is [Roorkee 1998] (a) a 2 l 2 + b 2 m 2 = n 2 (b) al 2 + bm 2 = n 2
(a) 0 (b) 1/2 (c) a 2 l + b 2 m = n (d) None of these
(c) 1 / 2 (d) 2 79. The locus of the point of intersection of mutually
x2 y2
69. For the ellipse 25 x 2 + 9 y 2 − 150 x − 90 y + 225 = 0 the perpendicular tangent to the ellipse 2 + 2 = 1 , is
eccentricity e = [Karnataka CET 2004]
a b
(a) 2/5 (b) 3/5 (a) A straight line (b) A parabola
(c) 4/5 (d) 1/5 (c) A circle (d) None of these
80. The equation of the tangent at the point (1/4, 1/4) of the
(x − 1) 2 (y + 1) 2
70. The eccentricity of the ellipse + = 1 is x2 y2
9 25 ellipse + = 1 is
4 12
[AMU 1999]
 Conic Sections 741

(a) 3 x + y = 48 (b) 3 x + y = 3  
(a) (b)
(c) 3 x + y = 16 (d) None of these 4 3
81. The angle between the pair of tangents drawn to the ellipse 3 2
(c) (d)
3 x 2 + 2 y 2 = 5 from the point (1, 2), is [MNR 1984] 4 3
90. The equation of the tangents drawn at the ends of the major
 12 
(a) tan −1   (b) tan −1 (6 5 ) axis of the ellipse 9 x 2 + 5 y 2 − 30 y = 0 , are
 5 
[MP PET 1999]
−1 
 12 
(c) tan   (d) tan −1 (12 5 ) (a) y = 3 (b) x =  5
 5 (c) y = 0, y = 6 (d) None of these
82. The equations of the tangents of the ellipse
9 x 2 + 16 y 2 = 144 which passes through the point (2, 3) is x2 y2
91. The equation of the normal to the ellipse 2
+ = 1 at the
[MP PET 1996] a b2
(a) y = 3, x + y = 5 (b) y = −3, x − y = 5 point (a cos  , b sin  ) is
(c) y = 4 , x + y = 3 (d) y = −4 , x − y = 3 ax by ax by
(a) − = a 2 − b 2 (b) − = a2 + b 2
x2 y2 sin  cos  sin  cos 
83. If any tangent to the ellipse + = 1 cuts off intercepts
a2 b2 (c)
ax
−
by
= a 2 − b 2 (d)
ax
−
by
= a2 + b 2
a2 b 2 cos  sin  cos  sin 
of length h and k on the axes, then = +
h k2 2
x2 y2
92. If the normal at the point P( ) to the ellipse + =1
(a) 0 (b) 1 14 5
(c) –1 (d) None of these intersects it again at the point Q(2 ) , then cos  is equal to
x2 y2
84. If the line y = mx + c touches the ellipse 2 + 2 = 1 , then 2 2
b a (a) (b) −
3 3
c= [MNR 1975; MP PET 1994, 95, 99]
3 3
(a)  b 2 m 2 + a 2 (b)  a 2 m 2 + b 2 (c) (d) −
2 2
(c)  b 2 m 2 − a 2 (d)  a 2 m 2 − b 2 x2 y2
2 2 93. The line y = mx + c is a normal to the ellipse 2
+ =1,
85. The ellipse
x
+
y
= 1 and the straight line y = mx + c a a2
a2 b 2 if c =
intersect in real points only if [MNR 1995] (a 2 + b 2 )m
(a) a 2 m 2  c 2 − b 2 (b) a 2 m 2  c 2 − b 2 (a) − (2 am + bm 2 ) (b)
a2 + b 2m 2
(c) a 2 m 2  c 2 − b 2 (d) c  b
x2 y2 (a 2 − b 2 )m (a 2 − b 2 )m
86. If y = mx + c is tangent on the ellipse + = 1 , then (c) − (d)
9 4 a2 + b 2m 2 a2 + b 2
the value of c is 94. The equation of normal at the point (0, 3) of the ellipse
(a) 0 (b) 3 / m 9 x 2 + 5 y 2 = 45 is [MP PET 1998]
(c)  9 m 2 + 4 (d)  3 1 + m 2 (a) y − 3 = 0 (b) y + 3 = 0
87. The locus of the point of intersection of the perpendicular (c) x-axis (d) y-axis
x2 y2 95. The equation of the normal at the point (2, 3) on the ellipse
tangents to the ellipse + = 1 is
9 4 9 x 2 + 16 y 2 = 180 , is [MP PET 2000]
[Karnataka CET 2003]
(a) 3 y = 8 x − 10 (b) 3 y − 8 x + 7 = 0
(a) x 2 + y 2 = 9 (b) x 2 + y 2 = 4
(c) 8 y + 3 x + 7 = 0 (d) 3 x + 2 y + 7 = 0
(c) x 2 + y 2 = 13 (d) x 2 + y 2 = 5
96. If the line x cos  + y sin  = p be normal to the ellipse
88. The eccentric angles of the extremities of latus recta of the
x2 y2 x2 y2
ellipse 2 + 2 = 1 are given by 2
+ = 1 , then [MP PET 2001]
a b a b2
−1  ae   be  (a) p 2 (a 2 cos 2  + b 2 sin 2  ) = a 2 − b 2
(a) tan    (b) tan −1   
 b   a  (b) p 2 (a 2 cos 2  + b 2 sin 2  ) = (a 2 − b 2 ) 2
 b   a  (c) p 2 (a 2 sec 2  + b 2 cosec 2 ) = a 2 − b 2
(c) tan −1    (d) tan −1   
 ae   be 
(d) p 2 (a 2 sec 2  + b 2 cosec 2 ) = (a 2 − b 2 )2
89. Eccentric angle of a point on the ellipse x 2 + 3 y 2 = 6 at a
97. The line lx + my + n = 0 is a normal to the ellipse
distance 2 units from the centre of the ellipse is
2 2
[WB JEE 1990] x y
+ = 1 , if [DCE 2000]
a2 b2
 742 Conic Sections
a2 b2 (a 2 − b 2 ) a2 b2 (a 2 − b 2 ) 2
(a) + = (b) + =
m2 l2 n2 l2 m2 n2
a 2
b 2
(a − b )
2 2 2
(c) 2
− 2
= (d) None of these
l m n2