Let L be common tangent line to the curves $4x^2 +9y^2=36 \text{ and } (2x)^2+(2y)^2=31$. Then the square of the slope of the line L is _________. Option: 1 3 Option: 2 4 Option: 3 5 Option: 4 6

$\\\text { Given curves are } \frac{\mathrm{x}^{2}}{9}+\frac{\mathrm{y}^{2}}{4}=1\;,\;x^{2}+y^{2}=\frac{31}{4}$

Let slope of common tangent be m

\begin{aligned} &\text { so tangents are } y=m x \pm \sqrt{9 m^{2}+4}\\ &y=m x \pm \frac{\sqrt{31}}{2} \sqrt{1+m^{2}} \end{aligned}

$\\\text { hence } 9 \mathrm{~m}^{2}+4=\frac{31}{4}\left(1+\mathrm{m}^{2}\right) \\ \Rightarrow 36 \mathrm{~m}^{2}+16=31+31 \mathrm{~m}^{2} \Rightarrow \mathrm{m}^{2}=3$

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