The locus of the center of a circle which touches the circle $$\left| {z - {z_1}} \right| = a\,\,{\text{and }}\left| {z - {z_2}} \right| = b$$      externally $$\left( {z,{z_1}\& {z_2}\,{\text{are complex numbers}}} \right)$$      will be

Solution :
Let the circle be $$\left| {z - {z_0}} \right| = r.$$  Then according to given conditions $$\left| {{z_0} - {z_1}} \right| = r + a\,\,{\text{and }}\left| {{z_0} - {z_2}} \right| = r + b.$$        Eliminating $$r,$$ we get $$\left| {{z_0} - {z_1}} \right| - \left| {{z_0} - {z_2}} \right| = a - b.$$
∴ Locus of center $${{z_0}}$$ is $$\left| {{z} - {z_1}} \right| - \left| {{z} - {z_2}} \right| = a - b,$$     which represents a hyperbola