Question

If a tangent of slope $2$ of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{1}=1$ passes through the point $\left(- 2,0\right)$ , then the value of $a^{2}$ is equal to

Answer 1

A tangent of slope $2$ to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{1}=1$ is
$y=2x\pm\sqrt{4 a^{2} + 1}$
Since, it passes through $\left(- 2,0\right)$
$\Rightarrow 0=2\left(- 2\right)\pm\sqrt{4 a^{2} + 1}$
$\Rightarrow 4a^{2}+1=16\Rightarrow a^{2}=\frac{15}{4}=3.75$