Let the circle be |z−z3|=r. Then according to given conditions |z3−z1|=r+a (Shown in the image) and |z3−z2|=r+b. (Shown in the image) Eliminating r, we get |z3−z1|−|z3−z2|=a−b ∴ Locus of center z3 is |z−z1|−|z−z2|=a−b= constant. Definition of hyperbola says, when difference of distance between two points is constant from a particular point then that particular point will lie on a hyperbola. Here distance of z1 from z3 is =r+a and distance of z2 from z3 is =r+b Now their difference =(r+a)−(r+b)=a−b= a constant ∴ Locus of z3 is a hyperbola.