The two conics a1x2+2h1xy+b1y2=c1 anda2x2+2h2xy+b2y2=c2 intersect in 4 concyclic points. Then

The equation of a conic passing through the intersection of the given conics isa1x2+2h1xy+b1y2−c1+λa2x2+2h2xy+b2y2−c2=0This equation will represent a circle, ifCoeff. of x2 = Coeff. of y2 and Coeff. of xy = 0⇒ a1+λu2=b1+λb2 and 2h1+2λh2=0Eliminating λ, from these two equations, we geta1−b1=h1h2a2−b2⇒a1−b1h2=a2−b2l1