# Two circles x2+y2=6 and x2+y2?6x+8=0 are given. Then find the equation of the circle through their point of intersection and the point (1,1).

$C_1:x^2+y^2=6\\* C_2:x^2+y^2-6x+8=0\\* C_3:C_1+\lambda C_2=0\\* \Rightarrow (x^2+y^2-6)+\lambda(x^2+y^2-6x+8)=0\\* \therefore C_3 \;passes\;through\;(1,1)\\*\Rightarrow (1+1-6)+\lambda(1+1-6+8)=0\\* \Rightarrow -4+\lambda(4)=0\Rightarrow \lambda=1\\*\Rightarrow \therefore C_3:(x^2+y^2-6)+(1)(x^2+y^2-6x+8)=0\\*\Rightarrow C_3:2x^2+2y^2-6x+2=0\\* \Rightarrow C_3:x^2+y^2-3x+1=0$

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