Let x 2 a 2 + y 2 b 2 = 1 , a > b be an ellipse, whose eccentricity is 1 2 and the length of the latus…

Given: Length of latus rectum of ellipse is $\sqrt{14}$.

$\Rightarrow \frac{2b^2}{a}=\sqrt{14} ...\left(i\right)$

Also, eccentricity is given as $\frac{1}{\sqrt{2}}$

$\Rightarrow \frac{1}{\sqrt{2}}=\sqrt{1-\frac{b^2}{a^2}}$

$\Rightarrow \frac{1}{2}=1-\frac{b^2}{a^2}$

$\Rightarrow \frac{1}{2}=1-\frac{a\sqrt{14}}{2a^2}$

$\Rightarrow \frac{-1}{2}=-\frac{\sqrt{14}}{2a}$

$\Rightarrow a=\sqrt{14}$

$\Rightarrow a^2=14, b^2=7$

Now, $e_H=\sqrt{1+\frac{b^2}{a^2}}$

$\Rightarrow e_H=\sqrt{1+\frac{1}{2}}$

$\Rightarrow e_H=\sqrt{\frac{3}{2}}$

$\Rightarrow {\left(e_H\right)}^2=\frac{3}{2}$