Question

The asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ form with any tangent to the hyperbola a triangle whose area is $a^2\tan \lambda$ in magnitude, then its eccentricity is :

• A. $\sec \lambda$
• B. $\cos ec\lambda$
• C. ${\sec }^2\lambda$
• D. $\cos e{c}^2\lambda$

Answer 1

The correct option is A $\sec \lambda$

Any tangent to hyperbola forms a triangle with the asymptotes which has constant area $ab$.

$\Rightarrow ab=a^2\tan \lambda$

$\Rightarrow \frac{b}{a}=\tan \lambda$

$e=\sqrt{1+\frac{b^2}{a^2}}$

$\Rightarrow e=\sqrt{1+{\tan }^2\lambda }=\sec \lambda$