Let $L$ be a common tengent 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

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Solution

$E : \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1, C : x^{2} + y^{2} = \frac{31}{4}$
equation of tangent to ellipse is
$y = m x \pm \sqrt{9 m^{2} + 4} . . . (1)$
equation of tangent to circle is
$y = m x \pm \sqrt{\frac{31}{4} m^{2} + \frac{31}{4}} . . . (2)$
Comparing equation $(1)$ & $(2)$
$9 m^{2} + 4 = \frac{31}{4} m^{2} + \frac{31}{44}$
$\Rightarrow 36 m^{2} + 16 = 31 m^{2} + 31$
$\Rightarrow 5 m^{2} = 15 \Rightarrow m^{2} = 3$

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