If m is the slope of a common tangent to the curves $\frac{x^2}{16}+\frac{y^2}{9}=1$ and $x^2+y^2=12$, then $12m^2$ is equal to :

Equation of the ellipse is

$\frac{x^2}{16}+\frac{y^2}{9}=1$

tangent equation is 

$$\begin{gathered} y=mx\pm \sqrt{a^2{m}^2+b^2} \\ y=mx\pm \sqrt{16m^2+9} ....\left(1\right) \\ {x}^2+y^2=12 \end{gathered}$$

equation of tangent to the circle is

$y=mx\pm \sqrt{12}\sqrt{1+m^2} ....\left(2\right)$

for common tangent equate eq. (1) and (2)

$$\begin{gathered} \Rightarrow 16m^2+9=12\left(1+m^2\right) \\ 16m^2-12m^2=3 \\ 4m^2=3 \\ 12m^2=9 \end{gathered}$$