Question

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

Answer 1

Given curves are $\frac{ x ^{2}}{9}+\frac{ y ^{2}}{4}=1$
$x^{2}+y^{2}=\frac{31}{4}$
let slope of common tangent be $m$
so tangents are $y=m x \pm \sqrt{9 m^{2}+4}$
$y=m x \pm \frac{\sqrt{31}}{2} \sqrt{1+m^{2}}$
hence $9 m ^{2}+4=\frac{31}{4}\left(1+ m ^{2}\right)$
$\Rightarrow 36 m ^{2}+16=31+31 m ^{2} \Rightarrow m ^{2}=3$