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Parabola
NIMCET 2024 (Date: 27.05.2024)
01. The equation of parabola whose focus is (5, 3) and directrix is 3x 4y 1 0 , is
(a) (4x 3y) 2 256x 142y 849 0 (b) (4x 3y) 2 256x 142y 849 0
(c) (3x 4y) 2 142x 256y 849 0 (d) (3x 4y) 2 256x 142y 849 0
02. The point on the parabola y 2 18x , for which the ordinate is three times the abscissa, is
(a) (6, 2) (b) (–2, –6) (c) (3, 18) (d) (2, 6)
03. The equation of the directrix of parabola 5y2 4x is
04.
05.
06.
(a) 4x 1 0
(a) 2/3
PS (b) 4x 1 0
(b) 1/3
Focus and directrix of the parabola x 2 8ay are
(c) 5x 1 0
(c) 4/3
(d) 5x 1 0
The point on the parabola y 2 8x . Whose distance from the focus is 8, has x-coordinate as
(a) 0 (b) 2 (c) 4 (d) 6
If the parabola y 2 4ax passes through (–3, 2), then length of its latus rectum is
(d) 4
(a) (0, –2a) and y = 2a (b) (0, 2a) and y = –2a (c) (2a, 0) and x = –2a (d) (–2a, 0) and x = 2a
07. The equation of the parabola with its vertex at the origin, axis on the y-axis and passing through the point
(6, –3) is
IN
(a) y 2 12x 6 (b) x 2 12y (c) x 2 12y (d) y 2 12x 6
08. Vertex of the parabola x 2 4x 2y 7 0 is
(a) ( 2,11 / 2) (b) ( 2, 2) (c) ( 2,11) (d) (2, 11)
09. The focus of the parabola 4y 2 6x 4y 5 is
(a) ( 8 / 5, 2) (b) ( 5 / 8, 1/ 2) (c) (1 / 2, 5 / 8) (d) (6 / 8, 1 / 2)
10. The equation of the directrix of the parabola y 2 4y 4x 2 0 is
3 3
(a) x = –1 (b) x = 1 (c) x (d) x
2 2
11. The line x – 1 = 0 is the directrix of the parabola y 2 kx 8 0 . Then one of the values of k is
(a) 1/8 (b) 8 (c) 4 (d) 1/4
12. Equation of the parabola with its vertex at (1, 1) and focus (3, 1) is
(a) (x 1)2 8(y 1) (b) (y 1)2 8(x 3) (c) (y 1)2 8(x 1) (d) (x 3)2 8(y 1)
13. x 2 t 2 , y 2t are the parametric equations of the parabola
(a) y 2 4x (b) y 2 4x (c) x 2 4y (d) y 2 4(x 2)
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2
14. The equation of a parabola is y 4x . P(1, 3) and Q(1, 1) are two points in the xy-plane. Then, for the
parabola
(a) P and Q are exterior points (b) P is an interior point while Q is an exterior point
(c) P and Q are interior points (d) P is an exterior point while Q is an interior point
15. The ends of a line segment are P(1, 3) and Q(1, 1). R is a point on the line segment PQ such that
PR : QR 1: . If R is an interior point of the parabola y 2 4x , then
3 1 3
(a) (0, 1) (b) , 1 (c) , (d) None of these
5 2 5
16. The straight line y 2x does not meet the parabola y 2 2x, if
1 1
(a) (b) (c) 4 (d) 1
4 4
17.
18.
19.
(a) x y 1 0
(a) y 3x 4 0
a a
PS
If the parabola y 2 4ax passes through the point (1, –2), then the tangent at this point
(b) x y 1 0
(b) 3y x 36 0
a a
(c) x y 1 0
(c) 3y x 36 0
a a
(d) x y 1 0
The equation of the tangent to the parabola y 2 16x , which is perpendicular to the line y 3x 7 is
(d) 3y x 36 0
If the tangent to the parabola y 2 ax makes an angle of 45o with x-axis, then the point of contact is
a a
(a) , (b) , (c) , (d) ,
2 2 4 4 2 4 4 2
20. The line x – y + 2 = 0 touches the parabola y 2 8x at the point
IN
(a) (2, –4) (b) (1, 2 2) (c) (4, 4 2) (d) (2, 4)
21. The equation of the tangent to the parabola at point (a / t 2 , 2a / t) is
(a) ty xt 2 a (b) ty x at 2 (c) y tx at 2 (d) y tx (a / t 2 )
22. Two tangents are drawn from the point (–2, –1) to the parabola y 2 4x. If is the angle between these
tangents, then tan =
(a) 3 (b) 1/3 (c) 2 (d) 1/2
1/3 1/3
a b 3
23. If , then the angle of intersection of the parabola y 2 4ax and x 2 4by at a
b
a
2
point other than the origin is
(a) / 4 (b) / 3 (c) / 2 (d) None of these
24. The equation of the common tangent touching the circle (x 3)2 y 2 9 and the parabola y 2 4x
above the x-axis , is
(a) 3y 3x 1 (b) 3y (x 3) (c) 3y x 3 (d) 3y (3x 1)
25. If a 0 and the line 2bx 3cy 4d 0 passes through the points of intersection of the parabolas
y 2 4ax and x 2 4ay then
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(a) d 2 (3b 2c) 2 0 (b) d 2 (3b 2c)2 0 (c) d 2 (2b 3c) 2 0 (d) d 2 (2b 3c)2 0
26. If x + y = k is a normal to the parabola y 2 12x, then k is
(a) 3 (b) 9 (c) –9 (d) –3
a
27. The equation of normal at the point , a to the parabola y 2 4ax , is
4
(a) 4x 8y 9a 0 (b) 4x 8y 9a 0 (c) 4x y a 0 (d) 4x y a 0
28. The point on the parabola y 2 8x at which the normal is parallel to the line x 2y 5 0 is
(a) ( 1 / 2, 2) (b) (1 / 2, 2) (c) (2, 1/ 2) (d) ( 2,1 / 2)
29. The equations of the normal at the ends of the latus rectum of the parabola y 2 4ax are given by
30.
31.
to each other, is
(a) x 2 2(y 6)
PQR lies on
PS
(a) x 2 y 2 6ax 9a 2 0
(c) x 2 y 2 6ay 9a 2 0
(b) x 2 2(y 6)
(b) x 2 y 2 6ax 6ay 9a 2 0
(d) None of these
The locus of the point of intersection of two normals to the parabola x 2 8y, which are at right angles
(c) x 2 2(y 6) (d) None of these
The normals at three points P, Q, R of the parabola y 2 4ax meet in (h, k), the centroid of triangle
(a) x = 0 (b) y = 0 (c) x = –a (d) y = a
32. If two of the three feet of normals drawn from a point to the parabola y 2 4x be (1, 2) and (1, – 2)
then the third foot is
IN
(a) (2, 2 2) (b) (2, 2 2) (c) (0, 0) (d) None of these
33. If the points (au 2 , 2au) and (av 2 , 2av) are the extremities of a focal chord of the parabola y 2 4ax, then
(a) uv 1 0 (b) uv 1 0 (c) u v 0 (d) u v 0
34. The locus of the midpoint of the line segment joining the focus to a moving point on the parabola y 2 4ax
is another parabola with the directrix
a a
(a) x a (b) x (c) x = 0 (d) x
2 2
35. The length of chord of contact of the tangents drawn from the point (2, 5) to the parabola y 2 8x, is
1 3
(a) 41 (b) 41 (c) 41 (d) 2 41
2 2
36. If b, k are the intercept of a focal chord of the parabola y 2 4ax , then K is equal to
ab b a ab
(a) (b) (c) (d)
ba ba ba ab
37. Equation of diameter of parabola y 2 x corresponding to the chord x – y + 1 = 0 is
(a) 2y = 3 (b) 2y = 1 (c) 2y = 5 (d) y = 1
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38. The length of the subtangent to the parabola y2 = 16x at the point whose abscissa is 4, is
(a) 2 (b) 4 (c) 8 (d) None of these
2
39. If P is point on the parabola y = 4ax such that the subtangent and subnormal at P are equal, then the
coordinates of P are
(a) (a, 2a) or (a, –2a) (b) (2a, 2 2a) or (2a, 2 2a)
(c) (4a, –4a) or (4a, 4a) (d) None of these
40. The pole of the line 2x = y with respect to the parabola y2 = 2x is
1 1 1
(a) 0, (b) , 0 (c) 0, (d) None of these
2 2 2
41. If the polar of a point with respect to the circle x 2 y 2 r 2 touches the parabola y 2 4ax, the locus
of the pole is
42.
2 r2
(a) y x
a
PS 2
(b) x
r 2
a
y 2 r2
(c) y x
a
2r2
(d) x y
a
A ray of light moving parallel to the x-axis gets reflected from a parabolic mirror whose equation is
(y 2)2 4(x 1). After reflection, the ray must pass through the point
(a) (0, 2) (b) (2, 0) (c) (0, –2) (d) (–1, 2)
IN
Answer Key
01. (a) 02. (d) 03. (c) 04. (d) 05. (c) 06. (a) 07. (c) 08. (a) 09. (b) 10. (d)
11. (c) 12. (c) 13. (d) 14. (d) 15. (a) 16. (b) 17. (c) 18. (a) 19. (d) 20. (d)
21. (a) 22. (a) 23. (b) 24. (c) 25. (d) 26. (b) 27. (b) 28. (b) 29. (a) 30. (a)
31. (b) 32. (c) 33. (b) 34. (c) 35. (c) 36. (a) 37. (b) 38. (c) 39. (a) 40. (a)
41. (a) 42. (a)
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