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Focal Chord

Let a, r, s, ...

Question

Let $a, r, s, t$ be nonzero real numbers. Let $P (a t^{2}, 2 a t), Q, R (a r^{2}, 2 a r) and S (a s^{2}, 2 a s)$ be distinct points on the parabola $y^{2} = 4 a x$ . Suppose that $P Q$ is the focal chord and lines $Q R$ and $P K$ are parallel, where $K$ is the point $(2 a, 0) .$

The value of $r$ is

A

- \frac{1}{t}

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B

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

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C

\frac{1}{t}

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D

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

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Solution

The correct option is D $\frac{t^{2} - 1}{t}$
We know that point $Q$ is,
$= (\frac{a}{t^{2}}, - \frac{2 a}{t})$

Since, $Q R$ is parallel to $P K$ , hence there slope must be same.
$∴ \frac{2 a t - 0}{a t^{2} - 2 a} = \frac{2 a r + \frac{2 a}{t}}{a r^{2} - \frac{a}{t^{2}}} \Rightarrow \frac{t}{t^{2} - 2} = \frac{r + \frac{1}{t}}{r^{2} - \frac{1}{t^{2}}} \Rightarrow \frac{t^{2} - 2}{t} = \frac{r^{2} - \frac{1}{t^{2}}}{r + \frac{1}{t}} \Rightarrow \frac{t^{2} - 2}{t} = r - \frac{1}{t} ∴ r = \frac{t^{2} - 1}{t}$

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