Let variable circle be |z−z0|=γ
Then, touches externally
|z0−z1|=a+γ and |z0−z2|=b+γ
eliminating r,
|z0−z1|−|z0−z1|=a−b−−(1)
also given,
|z1−z2|>|a−b|−−(2) [if |z1−z2|=a−b locus is a line with z lying on line segment joining z1 & z2 for |z1−z2|<|a−b| no locus substituting |a−b|=|z0−z1|−|z0−z1|
⇒|z1−z2|+|z0−z1|<|z0−z1|
conradicts property of trangle that sum of two sides > third side]
Thus (1)and (2) ensure locus is hyperbola