Show that the point of intersection of two perpendicular tangents to the hyperbola lies on the circle x² + y² = a² – b² and the write the definition of director circle.

The equation of any tangent to the hyperbola is $\mathrm{y}=\mathrm{mx}\pm \sqrt{{\mathrm{a}}^2{\mathrm{m}}^2-{\mathrm{b}}^2}$
Suppose P(x₁, y₁) is the point of intersection of two perpendicular tangents.
P lies on tangent $\Rightarrow {\mathrm{y}}_1={\mathrm{mx}}_1\pm \sqrt{{\mathrm{a}}^2{\mathrm{m}}^2-{\mathrm{b}}^2}$
$\begin{array}{l}\Rightarrow {\left({\mathrm{y}}_1-{\mathrm{mx}}_1\right)}^2={\mathrm{a}}^2{\mathrm{m}}^2-{\mathrm{b}}^2\\ \Rightarrow \left({\mathrm{x}}_1^2-{\mathrm{a}}^2\right){\mathrm{m}}^2-2{\mathrm{x}}_1{\mathrm{y}}_1\mathrm{m}+\left({\mathrm{y}}_1^2+{\mathrm{b}}^2\right)=0\end{array}$
This is quadratic equation in m and let m₁, m₂ be the roots.
$\begin{array}{l}\Rightarrow {\mathrm{m}}_1{\mathrm{m}}_2=\frac{{\mathrm{y}}_1^2+{\mathrm{b}}^2}{{\mathrm{x}}_1^2-{\mathrm{a}}^2}\Rightarrow \frac{{\mathrm{y}}_1^2+{\mathrm{b}}^2}{{\mathrm{x}}_1^2-{\mathrm{a}}^2}=-1\\ \Rightarrow {\mathrm{x}}_1^2+{\mathrm{y}}_1^2={\mathrm{a}}^2-{\mathrm{b}}^2\end{array}$
locus of P (x₁, y₁) is ${\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{a}}^2-{\mathrm{b}}^2$