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Condition for Two Lines to Be Perpendicular

The normal dr...

Question

The normal drawn at the point $P (a t_{1}^{2}, 2 a t_{1})$ on the parabola meets the curve again at $Q (a t_{2}^{2}, 2 a t_{2}) .$ then $t_{2} = ?$

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 A $t_{2} = - t_{1} - \frac{2}{t_{1}}$
The normal at $t_{1}$ is easily found to be $y = - t_{1} x + 2 a t_{1} + a t_{1}^{3}$ It passes through the point $(a t_{2}^{2}, 2 a t_{2})$
$∴ 2 a t_{2} = - t_{1} a t_{2}^{2} + 2 a t_{1} + a t_{1}^{3}$ or $2 a (t_{2} - t_{1}) = - a t_{1} (t_{2}^{2} - t_{1}^{2})$
$∴ 2 = - t_{1} (t_{2} + t_{1}) ∵ t_{2} \neq t_{1}$
$∴ t_{2} = - t_{1} - \frac{2}{t_{1}}$
Ans: A

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