Question

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is :

• A. $\frac{4}{3}$
• B. $\frac{4}{\sqrt{3}}$
• C. $\frac{2}{\sqrt{3}}$
• D. $\sqrt{3}$

Answer 1

Answer: (C)

$\ell =\frac{2b^2}{a}=8$
$\Rightarrow b^2=4a$ ....(1)
$2b=\frac{1}{2}(2ae)$
$2b=ae$
$2b=ae$ ....(2)
Squaring eqn. (2), we get
$4b^2=a^2{e}^2$
$\Rightarrow 4\frac{b^2}{a^2}=e^2$ and
we know that $e^2=1+\frac{b^2}{a^2}$
$\Rightarrow \frac{b^2}{a^2}=e^2-1$
$4(e^2-1)=e^2$
$4e^2-e^2=4$
$3e^2=4$
$e=\frac{2}{\sqrt{3}}$