We have equation of line y = x and equation of circle $x^2+y^2-2x=0$

Now, intersecting points of given line and circle
$x^2+x^2-2x=0$
$\Rightarrow 2x^2-2x=0$
$\Rightarrow 2x\left(x-1\right)=0$
$\Rightarrow x=\mathrm{0,1}$
When x = 0 then y =0 when x =1 then y = 1
$\therefore$ Coordinates of end points of diameter AB are (0, 0) and (1, 1)
$\therefore$ Required equation of circle with diameter AB
$\left(x-0\right)\left(x-1\right)+\left(y-0\right)\left(y-1\right)=0$
$\Rightarrow x^2-2+y^2-y=0$
$\Rightarrow x^2+y^2-x-y=0$