Let $a, s, t$ be nonzero real numbers. Let $P (a t^{2}, 2 a t), and S (a s^{2}, 2 a s)$ be distinct points on the parabola $y^{2} = 4 a x$ . If $s t = 1$ , then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

\frac{(t^{2} + 1)^{2}}{2 t^{3}}

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\frac{a (t^{2} + 1)^{2}}{2 t^{3}}

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\frac{a (t^{2} + 1)^{2}}{t^{3}}

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\frac{a (t^{2} + 2)^{2}}{t^{3}}

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Solution

The correct option is B $\frac{a (t^{2} + 1)^{2}}{2 t^{3}}$
Normal at $S (a s^{2}, 2 a s)$
$y = a s^{3} + 2 a s - s x y - \frac{a}{t^{3}} - \frac{2 a}{t} = \frac{- x}{t} \dots (1)$

Tangent at $P (a t^{2}, 2 a t)$
$t y = x + a t^{2} \Rightarrow y - a t = \frac{x}{t} \dots (2)$

From equation $(1)$ and $(2)$ ,
$y - a t = - y + \frac{a}{t^{3}} + \frac{2 a}{t} \Rightarrow 2 y = a (t + \frac{2}{t} + \frac{1}{t^{3}}) \Rightarrow y = \frac{a (t^{4} + 2 t^{2} + 1)}{2 t^{3}} = \frac{a (t^{2} + 1)^{2}}{2 t^{3}}$

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Let $a, r, s$ and $t$ be non-zero 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 point $(2 a, 0)$ .
If $s t = 1$ , then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is