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

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

Let the hyperbola be $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
We have , $\frac{2b^2}{a}=8$ and $2b=ae$
$\Rightarrow b^2=4aand2b=ae$
Consider, $2b=ae\Rightarrow 4b^2=a^2{e}^2\Rightarrow 4a^2(e^2-1)=a^2{e}^2\Rightarrow 4e^2-4=e^2\Rightarrow 3e^2=4e=\frac{2}{\sqrt{3}}$