Question

Find the locus of the mid-points of the chord of the parabola y2 = 4ax which subtend a right angle at the vertex.

Solution

Correct option is

y2 = 2a(– 4a).

Let P be t1 and Q be t2 and since PQ subtends a right angle at the vertex O(0, 0) therefore as in t­1 t­2 = – 4. If (h, k) be the mid-point, then 2h = (t12 + t22) and 2k = 2a(t1 + t2)

or   2h = a[(t1 + t2)2 – 2t1t2]

or  

  Locus is y2 = 2a(– 4a).

SIMILAR QUESTIONS

Q1

A tangent and a normal are drawn at the point P(16, 16) of the parabola y2 = 16x which cut the axis of the parabola at the points A and B respectively. If the center of the circle through P, A and B is C, then angle between PC and axis of x is

Q2

If x + y = k is normal to y2 = 12x, then k is

Q3

A circle drawn on any focal chord AB of the parabola y2 = 4axas diameter cuts the parabola again at and D. If the parameters of the points A, B, C, D be t1, t2 t3 and t1 respectively, then the value of t3 t4 is

Q4

The length of normal chord which subtends an angle of 900 at the vertex of the parabola y2 = 4x is

Q5

A focal chord of parabola y2 = 4x is inclined at an angle of  with the +ive direction of x-axis, then the slope of normal drawn at the ends of focal chord will satisfy the equation

Q6

If two different tangents of y2 = 4x are the normal’s to the parabola x2 = 4ay, then

Q7

If the parabola C and D intersect at a point A on the line L1, then equation of the tangent point L at A to the parabola D is

Q8

If a > 0, the angle subtended by the chord AB at the vertex of the parabola is

Q9

P is a point on the circle C, the perpendicular PQ to the major axis of the ellipse E meets the ellipse at M, then  is equal to

Q10

Equation of the diameter of the ellipse conjugate to the diameter respected by L is