If the length of the latus rectum of an ellipse is 4 units and the...

Given length of latus-rectum of an ellipse is $\dfrac{2{b}^{2}}{a} = 4 ⇒ {b}^{2} = 2a and$a - ae = \dfrac{3}{2} ⇒ a\left(1 - e\right) = \dfrac{3}{2}$.</p><p>Now we know,${b}^{2} = {a}^{2}\left(1 - {e}^{2}\right)$</p><p>$⇒ {b}^{2} = a \left(1 - e\right)a \left(1 + e\right)

$⇒ 2a =$\dfrac{3}{2}$a \left(1 + e\right) \left(∵ a \left(1 - e\right) = \dfrac{3}{2}\right)$

$⇒ \dfrac{4}{3} =$1 + e$</p><p>$⇒ e = $\dfrac{1}{3}$.