Two circles touch each other externally. The locus of the centre of a circle touching both these circles is

$$\begin{gathered} Let C_1 and C_2 be the centres of two circles of radii r_1 and r_2 respectively. \\ Let P be the centre and r be the radius of a circle touching the two given circles externally, \\ then \end{gathered}$$ $$\begin{gathered} {\mathrm{C}}_1\mathrm{P}=\mathrm{r}+{\mathrm{r}}_1 \\ {\mathrm{C}}_2\mathrm{P}=\mathrm{r}+{\mathrm{r}}_2 \\ {\mathrm{C}}_1\mathrm{P}-{\mathrm{C}}_2\mathrm{P}={\mathrm{r}}_1-{\mathrm{r}}_2=some cons\tan t \end{gathered}$$

point  P moves in such a way that the difference of its distances 
from two fixed points $C_1$​ and $C_2$ is constant. 
Therefore, P describes a hyperbola whose two foci are at $C_1$
and  $C_2$