Question

The asymptote 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. $\operatorname{cosec} \lambda$
• C. $\sec ^2 \lambda$
• D. $\operatorname{cosec}^2 \lambda$

Answer 1

Answer: (A)

$A = ab = a ^2 \tan \lambda \Rightarrow b / a =\tan \lambda$, hence $e ^2=1+\left( b ^2 / a ^2\right) \Rightarrow e ^2=1+\tan ^2 \lambda \Rightarrow e =\sec \lambda$