Let A and B are the given two circles with radii R1 and R2 respectively and their centres are F and G respectively.
Let C and D are variable circles so that each circle meet the given circles A and B externally.
Let r1 be the radius of circle C and P be its centre. Let r2 be the radius of circle D and Q be its centre.
PF = R1+r1 , PG = R2+r1 , PF-PG = R1 - R2
QF = R1+r2 , QG = R2+r2 , QF-QG = R1 - R2
Hence locus of centres of touching circles is set of points so that the difference between the distances from two given
fixed points ( here F and G ) to the point in locus is constant.
This locus is hyperbola and the fixed points F and G (Centers of given circles ) are foci of hyperbola