Find the orthogonal trajectories of the family of parabolas $y^{2} = 4 a x$ , where $a$ is a parameter.

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$\frac{y^{3}}{2} + x^{3} = c$

$\frac{2 y^{2}}{2} + 2 x = c$

$\frac{2 y^{2}}{2} + x^{2} = c$

$\frac{y^{2}}{2} + x^{2} = c$

answer is D.

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Detailed Solution

The given curve is,

$\begin{array}{l} y^{2} = 4 a x \\ \Rightarrow 2 y \frac{d y}{d x} = 4 a \\ \Rightarrow a = \frac{y}{2} \frac{d y}{d x} \end{array}$

Eliminating a from Eqs (i) and (ii), we get

$y^{2} = 4 x (\frac{y}{2} \frac{d y}{d x})$

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

Replacing $d y / d x by - d x / d y$ , we get

$y = 2 x \times - \frac{d y}{d x}$

$\Rightarrow y d y + 2 x d x = 0$

Integrating, we get

$\frac{y^{2}}{2} + x^{2} = c$

Which is the required orthogonal trajectory.

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