Let $y=m x+c$ be a common tangent to $9 x^{2}-16 y^{2}=144$ and $x^{2}+y^{2}=9 .$
Since, $y=m x+c \quad$ is a tangent to $9 x^{2}-16 y^{2}=144$
$\therefore c^{2}=a^{2} m^{2}-b^{2} \Rightarrow c^{2}=16 m^{2}-9 \quad \ldots$ (i)
Now, $y=m x+c$ is a tangent to $x^{2}+y^{2}=9$ $\therefore \frac{c}{\sqrt{m^{2}+1}}=3 \Rightarrow c^{2}=9\left(1+m^{2}\right)$
From Eqs. (i) and (ii), we get $16 m^{2}-9=9+9 m^{2}$
$\Rightarrow \quad m=\pm 3 \sqrt{\frac{2}{7}}$
On putting the value of $m$ in $\mathrm{Eq}$ (ii), we get $c=\pm \frac{15}{\sqrt{7}}$
Hence, $y=3 \sqrt{\frac{2}{7}} x+\frac{15}{\sqrt{7}}$ is a common tangent.