Question

If $\theta$ is the angle between the asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ with eccentricity $e$, then $\sec \frac{\theta }{2}$can be

• A. $e$
• B. $\frac{e}{2}$
• C. $\frac{e}{3}$
• D. $\frac{e}{\sqrt{e^2-1}}$

Answer 1

Answer: (A), (D)

The correct options are
A $e$
D $\frac{e}{\sqrt{e^2-1}}$

$\tan \frac{\theta }{2}=\frac{b}{a}\Rightarrow e^2-1={\tan }^2\frac{\theta }{2}\Rightarrow \sec \frac{\theta }{2}=e$
or $e^2-1={\cot }^2\frac{\theta }{2}\Rightarrow co\sec \frac{\theta }{2}=e$
$\Rightarrow \sec \frac{\theta }{2}=\frac{e}{\sqrt{e^2-1}}$.