Let L be a common tangent line to the curves
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) $9 m^{2}+4=\frac{31}{4} m^{2}+\frac{31}{4}$

$\Rightarrow 36 m^{2}+16=31 m^{2}+31$

$\Rightarrow 5 m^{2}=15$

$\Rightarrow m^{2}=3$