Question

If a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ having slope $2$ is normal to the circle $x^2+y^2+4x+1=0,$ then the maximum value of $ab$ is

Answer 1

Equation of tangent is $y=2x\pm \sqrt{4a^2+b^2}$

If this tangent is normal to the circle $x^2+y^2+4x+1=0,$ then it will pass through the centre $(-2,0)$ of the circle.
$\Rightarrow 0=-4\pm \sqrt{4a^2+b^2}\Rightarrow 4a^2+b^2=16$

Using $A.M.\ge G.M.$ inequality, we get
$\frac{4a^2+b^2}{2}\ge \sqrt{4a^2\times b^2}\Rightarrow ab\le 4$