If OB is the semi-minor axis of an ellipse, F _ 1 and F _ 2 are its...

In standard ellipse $\dfrac{{x}^{2}}{{a}^{2}} + \dfrac{{y}^{2}}{{b}^{2}} = 1$

Semi-major axis $OA = a$

Semi-minor axis $OB = b$

Focus ${F}_{1} ≡ \left(- ae, 0\right)$

${F}_{2} ≡ \left(ae, 0\right)$

$∠ {F}_{1}B{F}_{2} = 90 °$

$O{F}_{1} = O{F}_{2} ⇒ ∠ {F}_{1}BO = ∠ {F}_{2}BO = 45 °$

$⇒ \dfrac{ae}{b} = \tan 45 ° = 1$

$⇒ b = ae$

Also, ${b}^{2} = {a}^{2}\left(1 - {e}^{2}\right)$

$⇒ {a}^{2}{e}^{2} = {a}^{2}\left(1 - {e}^{2}\right)$

$⇒ 2{e}^{2} = 1$

$⇒ {e}^{2} = \dfrac{1}{2}$