Question

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

• A. $\frac{1 - e}{e}$
• B. $\frac{1}{e} - 1$
• C. $\frac{1}{e}$
• D. None of these

Answer 1

Answer: (C)

$\frac{1}{e}$

Answer 2

$b^{2} = a^{2}\left( e^{2} - 1 \right)$……………..(1)

since angle between asymptotes

= $2\tan^{- 1}\left( \frac{b}{a} \right)$ or$\theta = 2\tan^{- 1}\left( \frac{b}{a} \right)$

⇒ $\tan\frac{\theta}{2} = \frac{b}{a}$

∴ $\cos\frac{\theta}{2} = \frac{a}{\sqrt{a^{2} + b^{2}}}$= $\frac{1}{\sqrt{1 + \left( \frac{b}{a} \right)^{2}}}$

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