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

The circle through (0, 0) and (1, 0) is $(x-0)(x-1)+(y-0)(y-0)+\lambda y=0$

or $x^2+y^2-x+\lambda y=0$

with centre $\left(\frac{1}{2},\frac{-\lambda }{2}\right)$ and radius $\frac{\sqrt{1+{\lambda }^2}}{2}$

The two circles touch internally if $C_1{C}_2=\left|r_1-r_2\right|$$C_1{C}_2=$ distance between centres $=\frac{\sqrt{1+{\lambda }^2}}{2}$

$\therefore \frac{\sqrt{1+{\lambda }^2}}{2}=3-\frac{\sqrt{1+{\lambda }^2}}{2}\Rightarrow 1+{\lambda }^2=9,\lambda =\pm 2\sqrt{2}$

$\therefore$ Centre is $\left(\frac{1}{2},\pm \sqrt{2}\right).$