The orthogonal trajectories of the family of curves y=C x^2, (C is an…

The equation of the given family of curves is

$y=C x^2$               .......(i)

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

$\frac{d y}{d x}=2 C x$           .....(ii)

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

$y=\left(\frac{1}{2 x} \frac{d y}{d x}\right) x^2 \Rightarrow 2 y=x \frac{d y}{d x}$           .......(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 $\frac{d y}{d x}$ by $-\frac{d x}{d y}$ in equation (iii).

Replacing $\frac{d y}{d x}$ by $\frac{-d x}{d y}$ in (iii), we obtain

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

On integrating, we obtain

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

This is the required family of orthogonal trajectories.