The eccentricity of an ellipse having centre at the origin, axes...

Equation of ellipse whose centre at $\left(0, 0\right)$ is

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

It passes through $( 4, - 1 )$

$⇒ \dfrac{16}{{a}^{2}} + \dfrac{1}{{b}^{2}} = 1$

$⇒ 16{b}^{2} + {a}^{2} = {a}^{2}{b}^{2} . . . \left(i\right)$

Again it passes through $( - 2, 2 )$

$⇒ \dfrac{4}{{a}^{2}} + \dfrac{4}{{b}^{2}} = 1$

$⇒ 4{b}^{2} + 4{a}^{2} = {a}^{2}{b}^{2} . . . \left(ii\right)$

From the equation \left(i\right)$and \left(ii\right)$, we get,

$16{b}^{2} + {a}^{2} = 4{a}^{2} + 4{b}^{2}$

$⇒ {a}^{2} = 4{b}^{2}$

Now, we have, ${e}^{2} = 1 - \dfrac{{b}^{2}}{{a}^{2}} = 1 - \dfrac{1}{4} = \dfrac{3}{4}$

$⇒ e = \dfrac{\sqrt{3}}{2}$.