The locus of the middle points of the normal chords of the rectangular hyperbola  $x^2-y^2=a^2$ is

Equation of the normal at $(a\sec \theta , a\tan \theta )$ to the hyperbola $x^2-y^2=a^2$ is

$\frac{ax}{\sec \theta }+\frac{ay}{\tan \theta }=a^2+a^2=2a^2 \left(1\right)$

Let $(h, k)$ be the middle point of this normal then its equation is

$hx-ky-a^2=h^2-k^2-a^2\left(T=S_1\right)$

$\Rightarrow hx-ky=h^2-k \left(2\right)$

Comparing (1) and (2) we get

$\begin{array}{l}h\sec \theta =-k\tan \theta =\frac{h^2-k^2}{2a}\\ \Rightarrow \sec \theta =\frac{h^2-k^2}{2ah},\tan \theta =\frac{h^2-k^2}{-2ak}\\ \text{ So }1={\sec }^2 \theta -{\tan }^2 \theta =\frac{{\left(h^2-k^2\right)}^2}{4a^2}\left(\frac{1}{h^2}-\frac{1}{k^2}\right)\\ \Rightarrow {\left(h^2-k^2\right)}^3+4a^2{h}^2{k}^2=0\\ \text{ Locus of }(h, k)\text{ is }{\left(y^2-x^2\right)}^3=4a^2{x}^2{y}^2\end{array}$