Set of points from which tangent lines to given hyperbola are perpendicular.

Question

Find set of points from which start two perpendicular tangent lines to hyperbola $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$

Tangent lines to a hyperbola are $y=mx+\sqrt{a^2m^2-b^2}$ and $y=mx-\sqrt{a^2m^2-b^2}$. Two lines $y=a_1x+b_1$, $y=a_2x+b_2$ are perpendicular if $a_1a_2=-1$.

What I want to achieve is circle $\{x^2+y^2=a^2-b^2\}$.

Answer 1

The answer to your question, after one sign-change, is here: The locus of two perpendicular tangents to a given ellipse.

I particularly admire @Blue's answer. although to apply it in this case will require that you parameterize the hyperbola with $\sinh$ and $\cosh$ rather than sines and cosines.