Equation of a family of curve is $y^{2} = a x^{3}$ , what is the differential equation of the family of its orthogonal trajectory?

\frac{d y}{d x} = - \frac{2 x^{2}}{3 y}

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\frac{d y}{d x} = - \frac{3 x^{2}}{2 y}

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\frac{d y}{d x} = - \frac{2 x}{3 y}

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\frac{d y}{d x} = - \frac{3 x}{2 y}

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Solution

The correct option is C $\frac{d y}{d x} = - \frac{2 x}{3 y}$
We have, $y^{2} = a x^{3} \dots (1)$

Differentiating w.r.t. $x$ , we get

$2 y \frac{d y}{d x} = 3 a x^{2} \dots (2)$

Eliminating $k$ , from (2) using (1), we get

$2 x \frac{d y}{d x} = 3 y$

For orthogonal trajectory, substituting $\frac{d y}{d x}$ with $- \frac{d x}{d y}$ in (2), we get

$- 2 x \frac{d x}{d y} = 3 y$
$⟹ \frac{d y}{d x} = - \frac{2 x}{3 y}$

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