Question

Which among the following option(s) is/are correct for hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $n$ is the number of points on the plane through which perpendicular tangents are drawn to it and $e$ is the eccentricity.

• A. If $n=1$, then $e=\sqrt{2}$
• B. If $n>1$, then $0<e<\sqrt{2}$
• C. If $n=0$, then $e>\sqrt{2}$
• D. If $n=0$, then $e>2$

Answer 1

Answer: (A), (C)

If $n=0$, then $e>\sqrt{2}$

Locus of point intersection of perpendicular tangents is the director circle $x^2+y^2=a^2-b^2$
Now, $e^2=1+\frac{b^2}{a^2}$
$\left(1\right)$ If $a^2>b^2$, then there are more than one points on the circle.
$\Rightarrow 1<e^2<2\Rightarrow 1<e<\sqrt{2}$

$\left(2\right)$ If $a^2<b^2$ , there doesn't exist any such point on the plane. $\Rightarrow e^2>2\Rightarrow e>\sqrt{2}$

$\left(3\right)$ If $a^2=b^2$ there is exactly one point i.e., centre of hyperbola.
$\Rightarrow e=\sqrt{2}$