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(A) Given the equation of the parabola is $y = ax^2$ ... $(1)$Differentiating both sides with respect to $x$,we get $\frac{dy}{dx} = 2ax$.From $(1)$,w

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

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(A) Given the equation of the parabola is $y = ax^2$ ... $(1)$Differentiating both sides with respect to $x$,we get $\frac{dy}{dx} = 2ax$.From $(1)$,we have $a = \frac{y}{x^2}$. Substituting this into the derivative,we get $\frac{dy}{dx} = 2 \left(\frac{y}{x^2}\right) x = \frac{2y}{x}$.The differential equation of the orthogonal trajectory is obtained by replacing $\frac{dy}{dx}$ with $-\frac{dx}{dy}$:$-\frac{dx}{dy} = \frac{2y}{x}$.Rearranging the terms,we get $x dx = -2y dy$,or $x dx + 2y dy = 0$.Integrating both sides,we get $\int x dx + \int 2y dy = C$,which gives $\frac{x^2}{2} + y^2 = C$.

The equation to the orthogonal trajectories of the system of parabolas $y = ax^2$ is

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