A variable chord is drawn through the origin to the circle $x^2+y^2-2ax=0$. The locus of the centre of the circle drawn on this chord as diameter, is

Let (h, k) be the coordinates of the centre of the circle of which the given chord is the diameter. Then, (h, k) is the mid point of the chord and so its equation is  

$\begin{array}{l}{S}^{\mathrm{\prime }}=T\\ \Rightarrow h^2+k^2-2ah=hx+ky-a(x+h)\\ \Rightarrow x(h-a)+ky=h^2+k^2-ah\end{array}$

If it passes through $(0, 0)$, then $h^2+k^2-ah=0$

Hence, the locus of $(h, k)$ is $x^2+y^2-ax=0$