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Let $P\left(a t^{2}, 2 a t\right), Q,$ R(ar $^{2}, 2ar)$ 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
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The correct answer is:
$\frac{t^{2}-1}{t}$
Here, coordinate of $Q$ will be $\left(\frac{a}{t^{2}}, \frac{-2 a}{t}\right)$
Slope of $Q R=\frac{2}{r-\frac{1}{t}}$
Slope of $P K=\frac{2 a t}{a t^{2}-2 a}=\frac{2 t}{t^{2}-2}$
since, slope of $Q R=$ slope of $P K$
$\therefore \quad \frac{2}{r-\frac{1}{t}}=\frac{2 t}{t^{2}-2}$
$\Rightarrow \quad r=\frac{t^{2}-1}{t}$
Slope of $Q R=\frac{2}{r-\frac{1}{t}}$
Slope of $P K=\frac{2 a t}{a t^{2}-2 a}=\frac{2 t}{t^{2}-2}$
since, slope of $Q R=$ slope of $P K$
$\therefore \quad \frac{2}{r-\frac{1}{t}}=\frac{2 t}{t^{2}-2}$
$\Rightarrow \quad r=\frac{t^{2}-1}{t}$
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