Question

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. If $n=1$ then $e=\sqrt{2}$
• B. If $n=0$ then $e<\sqrt{2}$
• C. If $n>1$ then $0<e<\sqrt{2}$
• D. None of these

Answer 1

Answer: (A)

If $n=1$ then $e=\sqrt{2}$

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Equation of eccentricity $e=\frac{c}{a}$

i.e. $e=\frac{\sqrt{a^2+b^2}}{a}$

Let $n$ be the number of points on the plane,

If $n=1$, then

$e=\frac{\sqrt{1+1}}{1}=\sqrt{2}$

$\Rightarrow$ If $n=1$, then $e=\sqrt{2}$