x² – y² = a²
$\Rightarrow 2\text{x}-2\text{y}\frac{\text{dy}}{\text{dx}}=0$
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x}}{\text{y}}$
Product of slope of $x^2-y^2=a^2$ and slope of orthogonal trajectory of $x^2-y^2=a^2$ is $-1$
$\therefore$ Slope of orthogonal trajectory of $x^2-y^2=a^2$ is
$\frac{dy}{dx}=-\frac{y}{x}$
$\int \frac{\text{dx}}{\text{x}}=\int -\frac{\text{dy}}{\text{y}}$
⇒ logx = – logy + logc
logxy = logc
⇒ xy = c
which is a rectangular hyperbola.