JEE Main 2021 (Online) 26th February Evening ShiftQuestion 1
Let $L$ be a common tangent line to the curves $4x^{2}+9y^{2}=36$ and $(2x)^{2}+(2y)^{2}=31$. Then the square of the slope of the line $L$ is $\_\_\_\_\_\_$ .
Ellipse
Answer$3$
Solution
Curves: $\frac{x^2}{9}+\frac{y^2}{4}=1$ and $x^2+y^2=\frac{31}{4}$. Tangent $y=mx+c$. Ellipse: $c^2=9m^2+4$. Circle: $c^2=\frac{31}{4}(1+m^2)$. $36m^2+16=31+31m^2\Rightarrow 5m^2=15\Rightarrow m^2=3$.