The orthogonal trajectories of the family of

 curves $y=C{x}^2,$(C is an arbitrary constant), is 

The equation of the given family of curves is 

$y=C{x}^2$              …(i)

Differentiating (i) w.r.t. x, we get  

$\frac{dy}{dx}=2Cx$            …(ii)

Eliminating C between (i) and (ii), we obtain 

$y=\left(\frac{1}{2x}\frac{dy}{dx}\right)x^2\Rightarrow 2y=x\frac{dy}{dx}$            ..(iii)

This is the differential equation of the family of curves given in (i). 

The differential equation of the orthogonal trajectories of (i) is 

obtained by replacing $\frac{dy}{dx}$by $$$-\frac{dx}{dy}$in equation (iii)

Replacing $\frac{dy}{dx}$by $\frac{-dx}{dy}$ in(iii), we obtain 

$2y=-x\frac{dx}{dy}\Rightarrow 2ydy=-xdx$

On integrating, we obtain  

$y^2=-\frac{x^2}{2}+C\Rightarrow x^2+2y^2=2C$

This is the required family of orthogonal trajectories.