The length of the minor axis (along y -axis) of an ellipse in the...

Here, $2b = \dfrac{4}{\sqrt{3}} ⇒ b = \dfrac{2}{\sqrt{3}}$

Equation of tangents $≡ y = mx \pm \sqrt{{a}^{2}{m}^{2} + {b}^{2}}$

Comparing with $≡ y = - \dfrac{x}{6} + \dfrac{4}{3}$

$m = - \dfrac{1}{6}$ and ${a}^{2}{m}^{2} + {b}^{2} = \dfrac{16}{9}$

$⇒ \dfrac{{a}^{2}}{36} + \dfrac{4}{3} = \dfrac{16}{9}$

$⇒ \dfrac{{a}^{2}}{36} = \dfrac{16}{9} - \dfrac{4}{3} = \dfrac{4}{9}$

$⇒ {a}^{2} = 16$

Now, $e = \sqrt{1 - \dfrac{{b}^{2}}{{a}^{2}}}$

$⇒ e = \sqrt{1 - \dfrac{4}{3 \times 16}} = \sqrt{\dfrac{11}{12}}$

$⇒ e = \dfrac{1}{2}\sqrt{\dfrac{11}{3}}$