Passage for Q. (388 and 389) : Let $a, r, s, t$ be non-zero real numbers. Let $P\left(a t^{2}, 2 a t

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Passage for Q. (388 and 389) : Let $a, r, s, t$ be non-zero real numbers. Let $P\left(a t^{2}, 2 a tight), Q, R\left(a r^{2}, 2 a right)$ and $S\left(a s^{2}, 2 a sight)$ 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\

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(d) : The slope of $P K=$ the slope of $Q R$ $\Rightarrow \frac{-\frac{2 a}{t}-2 a r}{\frac{a}{t^{2}}-a r^{2}}=\frac{2 a t-0}{a t^{2}-2 a}\left(\because m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}ight)$\ $\Rightarrow \frac{t}{t^{2}-2}=-\frac{\frac{1}{t}+r}{\left(\frac{1}{t^{2}}-r^{2}ight)}=-\frac{1}{\frac{1}{t}-r}$\ $\Rightarrow \quad 1-t r=2-t^{2} \Rightarrow t r=t^{2}-1$\ $	herefore r=\frac{t^{2}-1}{t}$