The intercept on the line y=x  by the circle x2+y2−2x=0  is AB . Equation of the circle with AB  as a diameter is

Equation of any circle passing through the point of intersection of x2+y2−2x=0  and y=x  is x2+y2−2x+λ(y−x)=0 Or x2+y2−(2+λ)x+λy=0 Its centre is (2+λ2, −λ2) ,For AB  to be the diameter of the required circle, the centre must lie on AB , i.e.,2+λ2=−λ2⇒ λ=−1 .Thus, equation of required circle isx2+y2−x−y=0 .