If 3 x+4 y=12 is a tangent

$3 x+4 y=12 \sqrt{2}$

$\Rightarrow \quad 4 y=-3 x+12 \sqrt{2}$

$\Rightarrow \quad y=-\frac{3}{4} x+3 \sqrt{2}$

Now, condition of tangency, $c^{2}=a^{2} m^{2}+b^{2}$

$\therefore \quad 18=a^{2} \cdot \frac{9}{16}+9 \quad \Rightarrow \quad a^{2} \cdot \frac{9}{16}=9$

$\Rightarrow a^{2}=16 \Rightarrow a=4$

Eccentricity $e=\sqrt{1-\frac{b^{2}}{a^{2}}}=\sqrt{1-\frac{9}{16}}=\frac{\sqrt{7}}{4}$

$\therefore \quad a e=\frac{\sqrt{7}}{4} \cdot 4=\sqrt{7}$

$\therefore \quad$ Focus are $(\pm \sqrt{7}, 0)$

$\therefore \quad$ Distance between foci of ellipse $=2 \sqrt{7}$