Question

A hyperbola has its centre at the origin, passes through the point $(4,2)$ and has transverse axis of length $4$ along the $x$-axis. Then the eccentricity of the hyperbola is :

• A. $\sqrt{3}$
• B. $\frac{2}{\sqrt{3}}$
• C. $2$
• D. $\frac{3}{2}$

Answer 1

The correct option is B $\frac{2}{\sqrt{3}}$

Given $2a=4$
$\Rightarrow a=2$
The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through $(4,2)$
So, $\frac{4^2}{2^2}-\frac{2^2}{b^2}=1$
$\Rightarrow b^2=\frac{4}{3}$
$\therefore c^2=a^2+b^2=2^2+\frac{4}{3}$
$\Rightarrow c^2=\frac{16}{3}$
$\Rightarrow e=\frac{c}{a}=\frac{\sqrt{\frac{16}{3}}}{2}=\frac{2}{\sqrt{3}}$