Thank you for reporting, we will resolve it shortly
Q.
If $3x+4y=12\sqrt{2}$ is a tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{9}=1$ for some $a\,\in\,R,$ then the distance between the foci of the ellipse is :
$3x+4y=12\sqrt{12}$ is tangent to $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{9}=1$
$c^{2}=m^{2}a^{2}+b^{2}$
$\Rightarrow a^{2}=16$
$e=\sqrt{1-\frac{9}{16}}=\frac{\sqrt{7}}{4}$
Distance between focii $= 2ae =2\sqrt{7}$