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The locus of ...

Question

The locus of point such that two tangents drawn from it to the parabola $y^{2} = 4 a x$ are such that the slope of one is double the other is

y^{2} = \frac{9}{2} a x

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y^{2} = \frac{9}{4} a x

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y^{2} = 9 a x

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x^{2} = 4 a y

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Solution

The correct option is D $y^{2} = \frac{9}{2} a x$
REF.Image.
Given
slope of PA = 2 $\times$ slope of AB
$\frac{2 a d_{1} - a t_{1} - a t^{2}}{a t_{1}^{2} - a d_{1} d_{2}} = 2 \times \frac{2 a t_{2} - a t_{1} - a t_{2}}{a t_{2}^{2} - a t_{1} t_{2}}$
$\Rightarrow \frac{a (t_{1} - t_{2})}{a t_{2} (t_{1} - t_{2})} = 2 \times \frac{a (t_{2} - t_{1})}{a t_{2} (t_{2} - t_{1})}$
$\Rightarrow \frac{1}{t_{1}} = \frac{2}{t_{2}}$
$\Rightarrow t_{2} = 2 t_{1}$
$∴ P (2 a t_{1}^{2}, 3 a t_{1})$
let $y = 3 a t_{1}, x = 2 a t_{1}^{2}$
$∴ y^{2} = \frac{9 a}{2} x$

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