The intercept on the line $y=x$ by the circle ${ x }^{ 2 }+{ y }^{ 2 }-2x=0$ is $AB$. Equation of the circle with $AB$ as a diameter is

Equation of any circle passing through the point of intersection of

${ x }^{ 2 }+{ y }^{ 2 }-2x=0$ and $y=x$ is

${ x }^{ 2 }+{ y }^{ 2 }-2x+\lambda \left( y-x \right) =0$ $\Rightarrow { x }^{ 2 }+{ y }^{ 2 }-\left( 2+\lambda \right) x=\lambda y=0.$

Its centre is $\displaystyle \left( \frac { 2+\lambda }{ 2 } ,\frac { -\lambda }{ 2 } \right) .$

For $AB$ to be the diameter of the required circle, the centre must lie on $AB,$

i.e., $\displaystyle \frac { 2+\lambda }{ 2 } =\frac { -\lambda }{ 2 } \Rightarrow \lambda =-1.$

Thus, equation of required circle is ${ x }^{ 2 }+{ y }^{ 2 }-x-y=0.$