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

Answer: 3.75

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