The locus of the point of intersection of perpendicular tangents to the parabola $y^{2} = 4 x$ is.

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$x = a$

$x + a = 0$

$y = a$

$y + a = 0$

answer is B.

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Detailed Solution

Let $y = m x + \frac{a}{m}$ be any tangent to $y^{2} = 4 a x$ .

Let $P (x_{1}, y_{1})$ be the point of intersection

$\Rightarrow y_{1} = m x_{1} + \frac{a}{m} \Rightarrow x_{1} m^{2} - x y_{1} m + a = 0$ which is a quadratic equation in m.

If $m_{1}, m_{2}$ are the slopes of the two tangents then $m_{1}, m_{2}$ are the roots of the equation.

Since, the tangents are perpendicular $m_{1} m_{2} = - 1 \Rightarrow \frac{a}{x_{1}} = - 1 \Rightarrow a = - x_{1} \Rightarrow x_{1} + a = 0$

$∴ Locus of P (x_{1}, y_{1}) is x + a = 0$

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