Question

If point $(4,3\sqrt{3})$ be a point on standard hyperbola corresponding to eccentric angle $\frac{\pi }{3}$ radians. Find the eccentricity of the hyperbola.

• A. $e=\frac{2}{\sqrt{13}}$
• B. $e=\frac{13}{2}$
• C. $e=\frac{\sqrt{13}}{2}$
• D. $e=\frac{\sqrt{13}}{4}$

Answer 1

The correct option is C $e=\frac{\sqrt{13}}{2}$

Given: $(4,3\sqrt{3})$ point lies on the standard hyperbola, eccentric angle $=\frac{\pi }{3}$ radians

To Find: Eccentricity of the hyperbola

Step - $1$: Find the values of $a$ and $b$

Step - $2$: Find the eccentricity of the hyperbola

Any point $P\left(\theta \right)$ on the hyperbola: $(a\sec \theta ,b\tan \theta )$

$\therefore 4=a\sec \frac{\pi }{3}$

$\Rightarrow 4=2a$

$\Rightarrow a=2$

Also, $3\sqrt{3}=b\tan \frac{\pi }{3}$

$\Rightarrow 3\sqrt{3}=b\sqrt{3}$

$\Rightarrow b=3$

$\therefore$ Eccentricity of the hyperbola $e=\sqrt{1+\frac{b^2}{a^2}}$

$\Rightarrow e=\sqrt{1+\frac{3^2}{2^2}}$

$\Rightarrow e=\sqrt{1+\frac{9}{4}}$

$\Rightarrow e=\frac{\sqrt{13}}{2}$