Practicing Success
The orthogonal trajectories of the family of curves $y=C x^2$, (C is an arbitrary constant), is |
$x^2+2 y^2=2 C$ $2 x^2+y^2=2 C$ $x^2+y^2=2 C$ $x^2-2 y^2=2 C$ |
$x^2+2 y^2=2 C$ |
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. |