Complex Numbers Part 1

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.