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The normal at...

Question

The normal at the point $(a t_{1}^{2}, 2 a t_{1})$ on the parabola meets the parabola again in the point $(a t_{2}^{2}, 2 a t_{2})$ , then

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

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

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

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

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Solution

The correct option is C $t_{2} = - t_{1} - \frac{2}{t_{1}}$
Equation of the normal at point $(a t_{1}^{2}, 2 a t_{1})$ on parabola is
$y = - t_{1} x + 2 a t_{1} + a t_{1}^{3}$
It also passes through $(a t_{2}^{2}, 2 a t_{2})$
So, $2 a t_{2} = - t_{1} (a t_{2}^{2}) + 2 a t_{1} + a t_{1}^{3}$
$⟹ 2 t_{2} - 2 t_{1} = - t_{1} (t_{2}^{2} - t_{1}^{2})$
$⟹ 2 (t_{2} - t_{1}) = - t_{1} (t_{2} - t_{1}) (t_{2} + t_{1})$
$⟹ t_{1} + t_{2} = \frac{- 2}{t_{1}}$
$⟹ t_{2} = - t_{1} - \frac{2}{t_{1}}$
Hence, option B is correct.

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