If the chord joining the points $(a t_{1}^{2}, 2 a t_{1})$ and $(a t_{2}^{2}, 2 a t_{2})$ of the parabola $y^{2} = 4 a x$ passes through the focus of the parabola, then

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Solution

The correct option is A

$\frac{(y - 2 a t_{2})}{(2 a t_{2} - 2 a t_{1})} = \frac{x - a t_{2}^{2}}{(a t_{2}^{2} - a t_{1}^{2})}$ ; As focus i.e., (a,0) lies on it,
$\Rightarrow \frac{- 2 a t_{2}}{2 a (t_{2} - t_{1})} = \frac{a (1 - t_{2}^{2})}{a (t_{2} - t_{1}) (t_{2} + t_{1})} \Rightarrow - t_{2} = \frac{(1 - t_{2}^{2})}{(t_{2} + t_{1})}$
$\Rightarrow - t_{2}^{2} - t_{1} t_{2} = 1 - t_{2}^{2} \Rightarrow t_{1} t_{2} = - 1$ .

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