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 = 4 a $ ....(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 $
$e = \frac{2}{\sqrt{3}}$