From the origin, chords are drawn to the circle $x^2+y^2-2y=0,$ then the locus of the middle points of these chords is

Let  $y=mx\_\_\_\_\left(1\right)$ is chord of the circle $x^2+y^2-2y=0\_\_\_\_\_\left(2\right)$

Solving (1) and (2) we have  $O\left(0,0\right),A\left[\frac{2m}{m^2+1},\frac{2m^2}{m^2+1}\right]$

Let  $P\left(x,y\right)$ be the mid point of the chord.

$\left(x,y\right)=$mid – point of OA; $\left(x,y\right)=\left[\frac{m}{m^2+1},\frac{m^2}{m^2+1}\right]$

$x^2+y^2=\frac{m^2+m^4}{{\left(m^2+1\right)}^2}=\frac{m^2\left(1+m^2\right)}{{\left(1+m^2\right)}^2}=\frac{m^2}{1+m^2}\Rightarrow x^2+y^2=y\Rightarrow x^2+y^2-y=0$