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

(a, b, c) Locus of point of intersection of perpendicular tangent is the director circle, $x^2 + y^2 = a^2 – b^{2 }$
$e^2=1+\frac{b^2}{a^2},$ If  $a^2 > b^2$, then there are infinite points (or more than 1) points on the circle  $\Rightarrow e^2<2\Rightarrow e<\sqrt{2}$
If  $a^2 < b^2,$ there doesn’t exist any point on the plane  $\Rightarrow e^2>2\Rightarrow e>\sqrt{2}$
If   $a^2 = b^2$, there is exactly one point
(centre  of the hyperbola)  $\Rightarrow e=\sqrt{2}$