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Q.

If a tangent to the parabola $y^{2} = 4 a x$ meets the x axis in T and the tangent at the vertex A in P and the rectangle TAPQ is completed then locus of Q is

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a

$y^{2} + a x = 0$

b

$y^{2} - a x = 0$

c

$x^{2} + a x y = 0$

d

$x^{2} + a y = 0$

answer is A.

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Detailed Solution

$\begin{array}{l} y t = x + a t^{2} \Rightarrow (x, y) = (- a t^{2}, a t) \\ \Rightarrow t = \frac{y}{a} \\ \Rightarrow x = - a t^{2} \Rightarrow x = - a . \frac{y^{2}}{a^{2}} \\ \Rightarrow x = \frac{- y^{2}}{a} \\ \Rightarrow y^{2} + x a = 0 \Rightarrow y^{2} + a x = 0 \end{array}$

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