The locus of point of intersection of two perpendicular tangents to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is 

$\text{ Let }P(h,k)\text{ be the point of intersection of two perpendicular tangents equation of }$

$\text{ pair of tangents is }{\mathrm{SS}}_{11}={\left(S_1\right)}^2$

$\begin{array}{r}\Rightarrow \left(\frac{x^2}{a^2}-\frac{y^2}{b^2}-1\right)\left(\frac{h^2}{a^2}-\frac{k^2}{b^2}-1\right)={\left(\frac{hx}{a^2}-\frac{ky}{b^2}-1\right)}^2\\ \Rightarrow \frac{x^2}{a^2}\left(-\frac{k^2}{b^2}-1\right)-\frac{y^2}{b^2}\left(\frac{h^2}{a^2}-1\right)+\dots \dots ..=0\end{array}$

$\text{ Since equation (i) represents two perpendicular lines }$

$\begin{array}{l}\therefore \frac{1}{a^2}\left(-\frac{k^2}{b^2}-1\right)-\frac{1}{b^2}\left(\frac{h^2}{a^2}-1\right)=0\\ \Rightarrow -k^2-b^2-h^2+a^2=0\end{array}$

$\Rightarrow \text{ locus is }{x}^2+y^2=a^2-b^2$