Given, system of family of circles is
x2+y2+2fy+λ(x2+y2+2gx+k)=0 ⇒(1+λ)x2+(1+λ)y2+2gλx+2fy+λk=0 ⇒x2+y2+2⋅()x+2()y+=0 For point circle, radius should be zero.
∴+= ⇒g2λ2+f2=k(λ+λ2) ⇒(g2−k)λ2−kλ+f2=0 . . . (i)
Let the quadratic Eq. (i) has roots
λ1 and
λ2, then the coordinates of point circles are
c1(,) and
c2(,) ∵ Lines
OC1 and
OC2 are perpendicular to each other. So,
×=−1 . . . (ii)
∵ Product of roots of Eq. (i), we get
λ1λ2= . . . (iii)
From Eqs. (ii) and (iii), we get
⇒g2=−g2+k ⇒2g2=k⇒g2=