Let L be a common tangent line to the curves 4x² + 9y² = 36 and (2x)² + (2y)² = 31. Then the square of the slope of the line L is_____________
 

Given : Ellipse is $\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1$

$$\begin{gathered} let \tan gent be y=mx\pm \sqrt{9m^2+4} \\ It is \tan gent to the circle x^2+y^2=\frac{31}{4}then \\ \Rightarrow \frac{\left|\sqrt{9m^2+4}\right|}{\sqrt{m^2}+1}=\frac{\sqrt{31}}{2} \\ \Rightarrow 4(9m^2+4)=31(m^2+1) \\ \Rightarrow 36m^2-31m^2=31-16 \\ \Rightarrow 5m^2=15 \\ \Rightarrow m^2=3 \end{gathered}$$