If a directrix of a hyperbola centered at the origin and passing...

Let equation of hyperbola is $\dfrac{{x}^{2}}{{a}^{2}} - \dfrac{{y}^{2}}{{b}^{2}} = 1$

$∴$ It passes through \left(4, - 2\sqrt{3}\right) ⇒ \dfrac{16}{{a}^{2}} - \dfrac{12}{{b}^{2}} = 1$</p><p>$⇒ 16 - 12 \times \dfrac{{a}^{2}}{{b}^{2}} = {a}^{2} . . . \left(1\right)$</p><p>Equation of directrix is$x = \dfrac{a}{e} = \dfrac{4}{\sqrt{5}}$, given in question.</p><p>$⇒ {a}^{2} = \dfrac{16}{5}{e}^{2} . . . \left(2\right)$</p><p>And we know that${b}^{2} = {a}^{2}\left({e}^{2} - 1\right)$</p><p>$⇒ \dfrac{{b}^{2}}{{a}^{2}} = {e}^{2} - 1 . . . \left(3\right)$</p><p>$∴$From \left(1\right), \left(2\right) \text{ and } \left(3\right)$

$16 - \dfrac{12}{{e}^{2} - 1} = \dfrac{16}{5}{e}^{2}$

$⇒ 16{e}^{2} - 16 - 12 = \dfrac{16{e}^{2}}{5}\left({e}^{2} - 1\right)$

$⇒ 4{e}^{4} - 24{e}^{2} + 35 = 0$