Question

What will be the value of acute angle between the asymptotes if the length of transverse axis of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\sqrt{3}$ times the length of conjugate axis?

• A. ${30}^{\circ }$
• B. ${45}^{\circ }$
• C. ${60}^{\circ }$
• D. ${50}^{\circ }$

Answer 1

The correct option is C ${60}^{\circ }$

Given: Length of transverse axis is $\sqrt{3}$ times the length of conjugate axis; Hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

To Find: Acute angle between the asymptotes

The acute angle $\theta$ between both the asymptotes is given by $\tan \theta =|\frac{2ab}{a^2-b^2}|$

It is given that, $a=\sqrt{3}b$.

Substituting the $a$ and $b$ value in the above formula.

$\tan \theta =|\frac{2ab}{a^2-b^2}|$

$\Rightarrow \tan \theta =|\frac{2\left(\sqrt{3}b\right)b}{(\sqrt{3}b{)}^2-b^2}|$

​​​​​​​$\Rightarrow \tan \theta =|\frac{2\sqrt{3}{b}^2}{3b^2-b^2}|$

​​​​​​​$\Rightarrow \tan \theta =|\frac{2\sqrt{3}{b}^2}{2b^2}|$

​​​​​​​$\Rightarrow \tan \theta =\sqrt{3}$

​​​​​​​$\Rightarrow \theta ={60}^{\circ }$