The locus of the centre of a circle which touches the circle $\left|\mathrm{z}-{\mathrm{z}}_1\right|=\mathrm{a}\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}\left|\mathrm{z}-{\mathrm{z}}_2\right|=\mathrm{b}$ externally (z ,z₁ and Z₂  are complex numbers) will be

Let the circle be $\left|\mathrm{z}-{\mathrm{z}}_0\right|=\mathrm{r}$ . Then according to given conditions $|{\mathrm{z}}_0-{\mathrm{z}}_1|=\mathrm{r}+\mathrm{a}$ and $|{\mathrm{z}}_0-{\mathrm{z}}_2|=\mathrm{r}+\mathrm{b}$ . Eliminating r, we get $|{\mathrm{z}}_0-{\mathrm{z}}_1|-|{\mathrm{z}}_0-{\mathrm{z}}_2|=\mathrm{a}-\mathrm{b}$ , which represents a hyperbola.