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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_____________

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3

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

$l e t \tan g e n t b e y = m x \pm \sqrt{9 m^{2} + 4} I t i s \tan g e n t t o t h e c i r c l e x^{2} + y^{2} = \frac{31}{4} t h e n \Rightarrow \frac{|\sqrt{9 m^{2} + 4}|}{\sqrt{m^{2}} + 1} = \frac{\sqrt{31}}{2} \Rightarrow 4 (9 m^{2} + 4) = 31 (m^{2} + 1) \Rightarrow 36 m^{2} - 31 m^{2} = 31 - 16 \Rightarrow 5 m^{2} = 15 \Rightarrow m^{2} = 3$

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