Let x ^ 2 a ^ 2 + y ^ 2 b ^ 2 = 1, a > b be an ellipse, whose...

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

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

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

$⇒ \dfrac{1}{\sqrt{2}} = \sqrt{1 - \dfrac{{b}^{2}}{{a}^{2}}}$

$⇒ \dfrac{1}{2} = 1 - \dfrac{{b}^{2}}{{a}^{2}}$

$⇒ \dfrac{1}{2} = 1 - \dfrac{a\sqrt{14}}{2{a}^{2}}$

$⇒ \dfrac{- 1}{2} = - \dfrac{\sqrt{14}}{2a}$

$⇒ a = \sqrt{14}$

$⇒ {a}^{2} = 14, {b}^{2} = 7$

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

$⇒ {e}_{H} = \sqrt{1 + \dfrac{1}{2}}$

$⇒ {e}_{H} = \sqrt{\dfrac{3}{2}}$

$⇒ {\left({e}_{H}\right)}^{2} = \dfrac{3}{2}$