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Equation of Normal at Given Point

A normal draw...

Question

A normal drawn to parabola $y^{2} = 4 a x$ meet the curve again at $Q$ such that angle subtended by $P Q$ at vertex is $90^{\circ}$ , then coordinates of $P$ can be

(8 a, 4 \sqrt{2} a)

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(8 a, 4 a)

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(2 a, - 2 \sqrt{2} a)

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(2 a, 2 \sqrt{2} a)

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Solution

The correct option is D $(2 a, 2 \sqrt{2} a)$
$t_{2} = - t_{1} - \frac{2}{t_{1}}$
Also, $\frac{2 a t_{1}}{a t_{1}^{2}} \times \frac{2 a t_{2}}{a t_{2}^{2}} = - 1$
or $t_{1} t_{2} = - 4$
$∴ \frac{- 4}{t_{1}} = - t_{1} - \frac{2}{t_{1}}$
$\Rightarrow t_{1}^{2} + 2 = 4 \Rightarrow t_{1} = \pm \sqrt{2}$
So, the point will be
$(a t_{1}, 2 a t_{1}^{2}) = (2 a, \pm 2 \sqrt{2} a)$ .

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