For hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$, let n be the number of points on the plane through which perpendicular tangents are drawn.

Locus is director circle which is  ${\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{a}}^2-{\mathrm{b}}^2$

If ${\mathrm{a}}^2>{\mathrm{b}}^2$, then there are infinite ( or more than 1) points on the circle i.e., ${\mathrm{e}}^2<2\text{ or }\mathrm{e}<\sqrt{2}$

If ${\mathrm{a}}^2<{\mathrm{b}}^2$, there does not exist any point on the plane i.e., ${\mathrm{e}}^2>2\text{ or }\mathrm{e}>\sqrt{2}$

If ${\mathrm{a}}^2={\mathrm{b}}^2$, there is exactly one point (centre of the hyperbola) i.e., $\mathrm{e}=\sqrt{2}$