let the two circles be :

S₁: x² + y² +2g₁x + 2f₁y + c₁ = 0

S₂ : x² + y² +2g₂x +2f₂y + c₂ = 0

now the equation of circle passing through their point of intersection is given by:

 S₁ +  S₂=0

now  (-g₁ , -f₁) satifies this equation .also (-g₂,-f₂) satisfies it.

u will get  on putting -g1,-f1  that :   

similarly on putting -g2,-f2 u will get : 

now both these values of lamda must be equal since they represent the same circle .

hence on equating we get the required condition as:

if u carefully observe the values of lamda u will find that for the first one it is ratio of radius² / (ength of tangent on S2 from -g1,-f1)²

and similarly u will find for the second one but sort of inverse of it.thus one of the cases when this will be true will be when the two circles cut each other orthoganally and have same radius.