The two circles which passes through (0, a) and (0,−a) and touch the line y=mx+c will intersect each other at right angle, if

Equation of family circles through (0, a) and (0,−a) is[x2+(y−a)(y+a)]+λx=0,λ∈R⇒x2+y2+λx−a2=0(λ2)2+a2=−mλ2+c1+m2⇒(1+m2)[λ24+a2]=(mλ2−c)2⇒(1+m2)[λ24+a2]=m2λ24−mcλ+c2⇒λ2+4mcλ+4a2(1+m2)−4c2=0∴λ1λ2=4[a2(1+m2)−c2] since λ is coefficient of x⇒g1g2=[a2(1+m2)−c2]since given the circles cut orthogonallyAnd g1g2+f1f2=c1+c22⇒a2(1+m2)−c2=−a2Hence, c2=a2(2+m2)