Question

Let $L$ be a common tangent line to the curves $4x^2+9y^2=36$ and ${\left(2x\right)}^2+{\left(2y\right)}^2=31$. Then the square of the slope of the line $L$ is ___________.

Answer 1

Comparing equation of tangent to ellipse to that of circle:

Equation of ellipse is

$E:\frac{x^2}{9}+\frac{y^2}{4}=1$

Equation of circle is

$C:x^2+y^2=\frac{31}{4}$

Equation of the tangent to the ellipse is

$y=mx\pm \sqrt{\left(9m^2+4\right)}$ ……………..(i)

Equation of the tangent to the circle is

$y=mx\pm \sqrt{\left(\frac{31}{4}{m}^2+\frac{31}{4}\right)}$ ……………….(ii)

Comparing equation (i) and (ii).

$$\begin{gathered} \Rightarrow 9m^2+4=\frac{31}{4}{m}^2+\frac{31}{4} \\ \Rightarrow 36m^2+16=31m^2+31 \\ \Rightarrow 5m^2=15 \\ \Rightarrow m^2=3 \end{gathered}$$

Hence, the correct answer is $3$