Single Correct Choice Type Parabola

Single Correct Choice Type

M1. Let x(t) = cos3t and y(t) = sin2 t defines a curve where t is a parameter. This curve passes through
1 3 
the point  ,  at some t  t 0   0,  . The slope of the curve at that point is
8 4  2

4 4 4 3
(A) (B)  (C)  (D) 
3 3 3 3 3
M2. Given the directrix and the tangent at the vertex the number of parabolas that be drawn is
(A) 0 (B) 1 (C) 2 (D) infinitely many
M3. 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


(A) 2, 2 2  (B) (0, 0) 
(C) 2, 2 2  (D) None of these

M4. Point on the line x – y = 3 which is neatest to the curve x 2 = 4y is

(A) (0,3) (B) (3,0) (C) (2, –1) (D) None of these
M5. The equations of common tangents to the parabola y2 = 16x and x2 + y2 = 8 are
(A) y = x + 4, y = – x – 4 (B) y = 2x + 4, 2y + x = 9
(C) y = x + 9, y = – x – 4 (D) None of these
M6. Find the equation of a parabola whose focus is (1, 2) and directrix is the line x + y = 0

(A) x 2  2xy  y 2  4x  8y  10  0 (B) x 2  2xy  y 2  4x  8y  10  0

(C) x 2  2xy  y 2  4x  8y  10  0 (D) None of these

M7. If the parabola y2 = 4 ax passes through the point (3,2) then the length of its latus rectum
is

2 4 1
(A) (B) (C) (D) 4
3 3 3
M8. Find the focus and the equation of the parabola whose vertex is (6, –3) and directrix is
3x – 5y + 1 = 0.
(A) 0 (B) 2 (C) 1 (D) 4
M9. Radius of the circle that passes through origin and touches the parabola y2 = 4ax at the point
(a, 2a) is (a > 0)

5 5 3
(A) a (B) 2 2a (C) a (D) a
2 2 2

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Single Correct Choice Type Parabola

M10. The length of the latus rectum of the parabola 25[(x  2)2  (y  4) 2 ]  (4x  3y  12)2 is

16 8 12
(A) (B) (C) (D) None of these
5 5 5

M11. If the line y = mx + 1 intersect the parabola y 2  4x in two distinct points then
(A) m < 1 (B) m > 1 (C) m = 1 (D) m < 2
M12. The points on the axis of the parabola 3y 2  4y  6x  8  0 from where 3 distinct normals can
be drawn is given by

 4 19  2 19  2 16  2 7
(A)  a,  ;a  (B)  a,   ;a  (C)  a,   ;a  (D)  a,   ;a 
 3 9  3 9  3 9  3 9
M13. If the normals drawn at the end points of a variable chord PQ of the parabola y2 = 4 ax intersect
at parabola, then the locus of the point of intersection of the tangent drawn at the points P and
Q is
(A) x + a = 0 (B) x – 2a = 0 (C) y2 – 4x + 6 = 0 (D) None of these
M14. The equation of directrix of the parabola x 2  4y  6x  k  0 is y + 1 = 0 then
(A) k = 16 (B) k = –17 (C) Focus is (3, –3) (D) vertex is (3, –3)
M15. 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


(A) 2, 2 2  (B) (0, 0) 
(C) 2, 2 2  (D) None of these
M16. If the locus of a point which divides a chord of slope 2 of the parabola y2 = 4x internally in the
 b
ratio 1 : 2 is a parabola with vertex (a, b) and length of latus rectum  , then  a   is
 9

12 30
(A) 0 (B) 4 (C) (D)
81 81
M17. The tangent at P(1, 2) to the parabola y 2  4 x meets the tangent at vertex at H. If S be the focus
of the parabola & A be the area of the circle circumscribing SHP, then [A] is (where [.] is
greatest integer function)
(A) 1 (B) 2 (C) 3 (D) 4
M18. The points of contact Q and R of tangents from the point P (2, 3) to the parabola y 2  4 x are

1 
(A) (9, 6) and (1, 2) (B) (4, 4) and (9, 6) (C) (9, 6) and  ,1 (D) (1, 2) and (4, 4)
4 

M19. If normals are drawn form a point P (h. k) to the parabola y 2  4ax then the sum of the inter-
cepts which the normals cut off from the axis of the parabola is
(A) (h + a) (B) 3 (h + a) (C) 2(h + a) (D) None of these

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Single Correct Choice Type Parabola

M20. From a point  sin ,cos   if three normals can be drawn to the parabola y 2  4ax then the
value of ‘a’ is
   1  1 
(A)  ,1 (B)   ,0  (C)  ,1 (D) None of these
2   2  2 
M21. If two tangents drawn from the point () to the parabola y 2  4x be such that the slope of
one tangent is double of the other then
2 2 2 2
(A)    (B)    (C) 2  9 2 (D) None of these
9 9
M22. A tangent to a parabola x 2  4ay meets the hyperbola xy  c 2 in two points P and Q. The locus
of middle point of PQ is
(A) ellipse (B) parabola (C) hyperbola (D) circle
M23. The locus of a point ‘P’ where the three normals drawn from it on the parabola y 2  4ax are
such that two of then make complementary angles with x-axis is
(A) y 2  a  x  3a  (B) y 2  a  x  2a  (C) y 2  a  x  a  (D) y 2  a  x  4a 
M24. From orthocentre of ABC tangents are drawn to the parabola, then the chords of contact thus
obtained are
a 
(A) concurrent at  ,0  (B) concurrent at (a, 0)
3 
(C) concurrent at (2a, 0) (D) None of these
M25. A circle drawn on any focal chord of the parabola y 2  4ax as diameter cuts the parabola at two
points ‘t’ and ‘ t’ (other then the extremity of focal chord), then
(A) t t = –1 (B) t t = 2 (C) t t = 3 (D) None of these
M26. Let there be two parabolas y 2  4ax and y 2  4bx (where a  b and a, b, > 0). Then the locus
of the middle points of the intercepts between the parabola made of the lines parallel to the
common axis is
8ab 8ab 4ab
(A) y  (B) x  (C) y 
2 2 2
x y x (D) None of these
ba ba ba
M27. Equation of normal to the curve y  x 2  6x  6 which is perpendicular to the straight line
joining the origin to the vertex of the parabola is
(A) 4x – 4y – 11 = 0 (B) 4x – 4y + 1 = 0 (C) 4x – 4y – 21 = 0 (D) 4x – 4y + 21 = 0
M28. The set of values of ‘a’ for which at least one tangent to the parabola y 2  16ax becomes
normal to the circle x 2  y 2  2ax  8ay  5a 2  0 is
(A) {1} (B)  (C) [2, 4] (D) R
M29. If two different tangents of the parabola y 2  4x are normals of the parabola x 2  4by , then
1 1 1 1
(A) | b |  (B) | b |  (C) | b | 
(D) | b | 
2 2 2 2 2 2
M30. If three normals are drawn from the point (6, 0)to the parabola y = 4ax of which two are
2

mutually perpendicular, then length of its latus rectum is

(A) 4 (B) 6 (C) 8 (D) 10
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Single Correct Choice Type Parabola
M31. If f : R  R given by f (x) = x + ax + bx + c is a bijection, then
5 3

20
(A) (a, b) is a point inside the parabola x 
2
y
9

20
(B) (a, b) is a point inside the parabola y 
2
x
9

20
(C) (a, b) is a point outside the parabola x 
2
y
9

20
(D) (a, b) is a point outside the parabola y 
2
x
9

x 2 y2
M32. The line x + y = b bisects two distinct chords to the ellipse 2  2  2 which pass through the
a b
point (a, –b) on it. Then
(A) a2 + 6ab > 7b2 (B) a2 + 7ab > 6b2 (C) b2 + 6ab > 7a2 (D) b2 + 6a2 > 7ab
M33. The angle subtended at the focus by the normal chord at the point  ,   ,    0  on the pa-
rabola x2 = 4y is

 1 1 1 1 
(A) (B) tan (C) tan (D)
4 2 4 2
M34. Normals are drawn from the point P to the parabola y2 = 4x. Let m1, m2, m3 be the slopes of the
normals. If the locus of P when m1m2 = a is part of the given parabola,then a is
(A) 1 (B) – 1 (C) – 2 (D) 2
M35. If the lines (y – 301) = m1 (x + 201) and (y – 301) = m2 (x + 201) are the tangents of the
parabola y 2  804x , then

(A) m1  m 2  0 (B) m1m 2  1 (C) m1m 2  1 (D) m1  m 2  1

M36. The locus of the foot of perpendicular from the focus of any tangent to the parabola y 2  4ax
is
(A) x = 0 (B) y = 0 (C) x = – a (D) None of these
M37. The line x + y = 6 is a normal to the parabola y = 8x at the point
2

(A) (18, –12) (B) (4, 2) (C) (2, 4) (D) (3, 3)

M38. If the distance between the foci and the distance between the two directrix of the hyperbola

x 2 y2
  1 are in the ratio 3 : 2, then b : a is (a, b > 0)
a 2 b2

(A) 1: 2 (B) 3: 2 (C) 1 : 2 (D) 2 : 1

M39. If (–2, 7) is the highest point on the graph of y  2x 2  4ax  k, then k equals

(A) 31 (B) 11 (C) –1 (D) – 1/3

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