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.
