The centre of a circle passing through the points $\left(0,0\right)$ , $\left(1,0\right)$  and touching the circle $x^2+y^2=9$  is

Circle passes through $\left(0,0\right)$  and $\left(1,0\right)$ , so let the equation of the circle be

$x^2+y^2-x+ky=0$  solving with $x^2+y^2=9$

We get $9-x+ky=0$

$\Rightarrow x=9+ky$

$\Rightarrow {\left(9+ky\right)}^2+y^2=9$

$\Rightarrow \left(k^2+1\right)y^2+18ky+72=0$

For equal roots ${\left(18k\right)}^2-4.72\left(k^2+1\right)=0$

$\Rightarrow k=\pm 2\sqrt{2}$

Hence the coordinates of the centre are $\left(\frac{1}{2},\sqrt{2}\right)$  or $\left(\frac{1}{2},-\sqrt{2}\right)$