The condition that the two circles x2+y2+2ax+c=0,x2+y2+2by+c=0  may touch each other is

C1C2=r1+r2 ⇒a2+b2=a2−c+b2−csquarring on both sidesa2+b2=a2-c+ b2-c+2(a2-c)(b2-c)c=(a2-c)(b2-c)squarring again on both sidesc2=a2b2-c(a2+b2)+c2⇒a2b2=c(a2+b2)⇒1a2+1b2=1c