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
a2=c2(2m+1)
a2=c2(2+m2)
c2=a2(2+m2)
c2=a2(2m+1)
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 orthogonally
And g1g2+f1f2=c1+c22
⇒a2(1+m2)−c2=−a2
Hence, c2=a2(2+m2)