Answer
$x^2-y^2 = K$
Work Step by Step
Given that, $y = \frac{c}{x}$ _______(1)
Differentiate with respect to $x$
\[\frac{dy}{dx} = \frac{-c}{x^2}\]
From (1)
$\frac{dy}{dx}=\frac{-(xy)}{x^2}=\frac{-y}{x} $ ______(2)
Replace $\frac{dy}{dx}$ by $\frac{-dx}{dy}$ in (2) [ For orthogonal trajectories]
$\frac{-dx}{dy}= \frac{-y}{x}$
$x dx = y dy$
Integrating, $\int x dx = \int y dy +k $
$\frac{x^2}{2} = \frac{y^2}{2}+k$
$x^2-y^2 = K$ , where $K=2k$
