From the figure it is clear that the intersection point of two direct common tangents lies on $x$-axis.

Also $\Delta P T_{1} C_{1} \sim \Delta P T_{2} C_{2}$

$\therefore \quad P C_{1}: P C_{2}=2: 1$

or $P$ divides $C_{1} C_{2}$ in the ratio $2: 1$ externally

$\therefore \quad$ Coordinates of $P$ are $(6,0)$.

Let the equation of tangent through $P$ be

$y=m(x-6)$

Asit touches $x^{2}+y^{2}=4$

$\therefore\left|\frac{6 m}{\sqrt{m^{2}+1}}\right|=2 \Rightarrow 36 m^{2}=4\left(m^{2}+1\right)$

$\therefore \quad m=\pm \frac{1}{2 \sqrt{2}}$

$\therefore \quad$ Equations of common tangents are

$y=\pm \frac{1}{2 \sqrt{2}}(x-6)$

Also $x=2$ is the common tangent to the two circles.