How many orthogonal trajectories for the given family of curves when 'a' is the parameter.
$y = a x^{2}$

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Solution

Find orthogonal trajectories for $y = a x^{2}$ $a$ is parameter.
$y = a x^{2}$
$\frac{d y}{d x} = 2 a x \frac{d y}{d x} = \frac{2 y}{x}$
For orthogonal trajectories change $\frac{d y}{d x} \Rightarrow - \frac{d x}{d y}$
$- \frac{d x}{d y} = \frac{2 y}{x} 2 \int y d y = - \int x d x \frac{2 y^{2}}{2} = - \frac{x^{2}}{2} + C 2 y^{2} + x^{2} = C$

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