A tangent P T is drawn to the circle x^ 2 +y^ 2 =4 at the point P(...

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.