Question

Let $\alpha$ be the angle in radians between $\frac{x^2}{36}+\frac{y^2}{4}=1$ and the circle $x^2+y^2=12$ at their points of intersection. If $\alpha =ta{n}^{-1}\frac{k}{2\sqrt{3}}$, then the value of $\frac{k^2}{4}$ is

Answer 1

The points of intersection,
$\Rightarrow y=\pm \sqrt{3}$ and $x=\pm 3$
consider the point $P(3,\sqrt{3})$
Equation of the tangent at P to the circle is $3x+\sqrt{3}y=12$
Equation of the tangent at P to the ellipse is $\frac{x}{12}+\frac{\sqrt{3}}{4}y=1$ if $\alpha$ is angle between these tangents, then $tan\alpha =\frac{2}{\sqrt{3}}$