The abscissa of point on curve $a y^{2} = x^{3},$ normal at which cuts off equal intercepts from the coordinate axes is

\frac{2 a}{9}

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\frac{4 a}{9}

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- \frac{4 a}{9}

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- \frac{2 a}{9}

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Solution

The correct option is D $\frac{4 a}{9}$
Slope of such normal is 1.
$\Rightarrow \frac{d y}{d x} = 1$
$a y^{2} = x^{3}$
$\Rightarrow 2 a y \frac{d y}{d x} = 3 x^{2}$
$\Rightarrow y = \frac{3 x^{2}}{2 a} \Rightarrow a {(\frac{3 x^{2}}{2 a})}^{2} = x^{3} \Rightarrow x = \frac{4 a}{9}$

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