A tangent of slope 2 of the ellipse x ^ 2 a ^ 2 + y ^ 2 1 = 1...

A tangent of slope $2$ to the ellipse $\dfrac{{x}^{2}}{{a}^{2}} + \dfrac{{y}^{2}}{1} = 1$ is
$y = 2x \pm \sqrt{4{a}^{2} + 1}$ which passes through \left(- 2, 0\right)$<br />$⇒ 0 = 2\left(- 2\right) \pm \sqrt{4{a}^{2} + 1}$<br />$⇒ 4{a}^{2} + 1 = 16 ⇒ {a}^{2} = \dfrac{15}{4}$<br />$⇒ {e}^{2} = 1 - \dfrac{4}{15} = \dfrac{11}{15}$<br />$⇒ 3{e}^{2} = 2.2