Question

The eccentricity of the hyperbola whose length of conjugate axis is equal to half of the distance between its foci is $e$, then $3e^2=$

Answer 1

Let equation of the hyperbola is
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
$\therefore$ length of conjugate axis will be $2b$ unit and foci will be $(\pm ae,0)$
Given
$2b=ae$
We know $b^2=a^2(e^2-1)$
$\Rightarrow \frac{e^2}{4}=e^2-1\Rightarrow 3e^2=4$