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
Solution:
$E: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \quad C: x^{2}+y^{2}=\frac{31}{4}$
equation of tangent to ellipse is $y=m x \pm \sqrt{9 m^{2}+4}$
equation of tangent to circle is
$y=m x \pm \sqrt{\frac{31}{4} m^{2}+\frac{31}{4}}$...(2)
Comparing equation (i) $\backslash$ (ii) $9 m^{2}+4=\frac{31}{4} m^{2}+\frac{31}{4}$
$\Rightarrow 36 m^{2}+16=31 m^{2}+31$
$\Rightarrow 5 m^{2}=15$
$\Rightarrow m^{2}=3$