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If the normal...

Question

If the normal at $t_{1}$ on the parabola $y^{2} = 4 a x$ meets it again at $t_{2}$ then $t_{2} =$

t_{1} + \frac{2}{t_{1}}

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- t_{1} - \frac{2}{t_{1}}

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t_{1} - \frac{2}{t_{1}}

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- t_{1} + \frac{2}{t_{1}}

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Solution

The correct option is C $- t_{1} - \frac{2}{t_{1}}$
The equation of normal to the parabola is $y + x t_{1} = 2 a t_{1} + a {t_{1}}^{3}$
Since, it passes through $t_{2}$
Therefore, $2 a t_{2} + a {t_{2}}^{2} t_{1} =$ $2 a t_{1} + a {t_{1}}^{3}$
$\Rightarrow t_{2} = - t_{1} - \frac{2}{t_{1}}$

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