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Focal Chord

Let P at 2, 2...

Question

Let $P (a t^{2}, 2 a t), Q, R (a r^{2}, 2 a r)$ be three points on a parabola $y^{2} = 4 a x .$ If $P Q$ is the focal chord and $P K, Q R$ are parallel where the co-ordinates of $K$ is $(2 a, 0),$ then the value of $r$ is

A

\frac{t}{1 - t^{2}}

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B

\frac{1 - t^{2}}{t}

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C

\frac{t^{2} + 1}{t}

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D

\frac{t^{2} - 1}{t}

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Solution

The correct option is D $\frac{t^{2} - 1}{t}$

Coordinates of $Q \equiv (\frac{a}{t^{2}}, \frac{- 2 a}{t})$
$P K$ and $Q R$ are parallel.
$\Rightarrow m_{P K} = m_{Q R}$
$\frac{2 a t - 0}{a t^{2} - 2 a} = \frac{2 a r + \frac{2 a}{t}}{a r^{2} - \frac{a}{t^{2}}}$

$\Rightarrow \frac{2 t}{t^{2} - 2} = \frac{2}{r - \frac{1}{t}}$
$\Rightarrow r = \frac{t^{2} - 1}{t}$

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