Question

Let $L$ be a common tangent line to the curves $4x^2+9y^2=36$ and $(2x{)}^2+(2y{)}^2=31.$ Then the square of the slope of the line $L$ is

Answer 1

$E:\frac{x^2}{9}+\frac{y^2}{4}=1,C:x^2+y^2=\frac{31}{4}$
equation of tangent to ellipse is
$y=mx\pm \sqrt{9m^2+4}...\left(1\right)$
equation of tangent to circle is
$y=mx\pm \sqrt{\frac{31}{4}{m}^2+\frac{31}{4}}...\left(2\right)$
Comparing equation $\left(1\right)$ & $\left(2\right)$
$\Rightarrow \pm \sqrt{9m^2+4}=\pm \sqrt{\frac{31}{4}{m}^2+\frac{31}{4}}$
$\Rightarrow 4(9m^2+4)=31m^2+31$
$\Rightarrow 36m^2+16=31m^2+31$
$\Rightarrow 5m^2=15\Rightarrow m^2=3$