The eccentricity of the hyperbola whose length of its conjugate...

For any standard hyperbola we know that

eccentricity, $e = \sqrt{1 + \dfrac{{b}^{2}}{{a}^{2}}}$, conjugate axis $= 2b$ and distance between the foci $= 2ae$.

Given, $2b = \dfrac{1}{2}\left(2ae\right)$.

$⇒$ $\dfrac{b}{a} = \dfrac{e}{2}$

Now, $e = \sqrt{1 + \dfrac{{b}^{2}}{{a}^{2}}} = \sqrt{1 + \dfrac{{e}^{2}}{4}}$

$⇒ {e}^{2} = 1 + \dfrac{{e}^{2}}{4}$

$⇒ \dfrac{3}{4}{e}^{2} = 1$

$⇒ {e}^{2} = \dfrac{4}{3}$

$⇒ e = \dfrac{2}{\sqrt{3}}$