Circles


Radical axis of both the circles is $s_{1}−s_{2}=x+y=0$
Taking any point on the radical axis as center, a unique circle with radius as length of tangent to any of the given two circles can be drawn which is orthogonal to both the circles. Let any point on $x+y=0$ be $(λ,−λ),$ so the radius of the circle $=λ_{2}+λ_{2}+6λ+5 $
So, the circle cutting both circles orthogonally will be $(x−λ)_{2}+(y+λ)_{2}=2λ_{2}+6λ+5$
l.e., $(x_{2}+y_{2}−5)−2λ(x−y+3)=0$
$∴$ All such circles pass through the points of intersection of $x_{2}+y_{2}=5$ and $x−y+3=0$
So, all circles pass through the two fixed points, viz (1,2) and (2,1)