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

Answer: 3

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 $y=mx\pm \sqrt{9m^2+4}$
$y=mx\pm \frac{\sqrt{31}}{2}\sqrt{1+m^2}$
hence $9m^2+4=\frac{31}{4}\left(1+m^2\right)$
$\Rightarrow 36m^2+16=31+31m^2\Rightarrow m^2=3$