Let $L$ be a common tangent line to the curves $4 x^{2}+9 y^{2}=36$ and $(2 x)^{2}+(2 y)^{2}=31$. Th

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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

Professor PC Thomas Classes · Answer

(3) : Given curves are\ $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ and $x^{2}+y^{2}=\frac{31}{4}$\ Let slope of common tangent be $m$.\ So, tangents of given ellipse and circle are respectively\ $y=m x \pm \sqrt{9 m^{2}+4}$\ and $y=m x \pm \frac{\sqrt{31}}{2} \sqrt{1+m^{2}}$\ Hence, $9 m^{2}+4=\frac{31}{4}\left(1+m^{2}ight)$\ $\Rightarrow \quad 36 m^{2}+16=31+31 m^{2} \Rightarrow 5 m^{2}=15$\ $\Rightarrow \quad m^{2}=3$\