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If a tangent to the parabola $y 2 = 4 a x$ meets the x− axis at T and the tangent at the vertex A in P and the rectangle TAPQ is completed. What is the locus of Q?

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y 2 + 4 a x = 0

y 2 + 2 a x = 0

y 2 = 2 a x

y 2 + a x = 0

answer is D.

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Detailed Solution

Let the diagram of the situation be,
seo

To find the other vertices of the triangle, as all parabolas of the type

y 2 = 4 a x

has the coordinates at their vertices at the origin of coordinate of A is

A (0, 0)

, and the tangent at the vertex is y-axis. As all the points on x-axis like T has their y-coordinate zero and also T lies on the tangent

t y = x + a t 2

, we have

t (0) = x + a t 2 x = − a t 2

So, the coordinate of T is

T (− a t 2, 0)

. As all the points on x-axis like P has their y-coordinate zero and also P lies on the tangent

t y = x + a t 2

, we have

t y = 0 + a t 2 y = a t

So, the coordinate of P is

P (0, a t)

.
Take the point of intersection of diagonals TP and AQ is M. As the midpoints of the diagonals QA and TP is the same, then M irrespective of motion of the point Q.
By using the midpoint formula, the coordinate of M is,

Midpoint of TP = Midpoint of AQ − a t 2 + 0 2, 0 + a t 2 = 0 + h 2, 0 + k 2 − a t 2 2, a t 2 = h 2, k 2 h 2 = − a t 2 2, k 2 = a t 2 h = − a t 2, k = a t

To find the value of t, and we have

k = a t t = k a

Substitute the value in h, and we have

h = − a t 2 = − a k a 2 h = − a k 2 a 2 k 2 + a h = 0

Replace the point

(h, k)

(x, y)

, and the locus of the point Q is

y 2 + a x = 0

.
Hence, the correct option is 4.

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