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(D) Let the required circle be $S_2$ with center $(h, k)$ and radius $r$. Since it passes through $(0, 0)$ and $(1, 0)$,the center must lie on the per

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$(1/2, \pm \sqrt{2})$

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(D) Let the required circle be $S_2$ with center $(h, k)$ and radius $r$. Since it passes through $(0, 0)$ and $(1, 0)$,the center must lie on the perpendicular bisector of the segment joining $(0, 0)$ and $(1, 0)$,which is the line $x = 1/2$. Thus,$h = 1/2$.Since it passes through $(0, 0)$,the radius $r = \sqrt{h^2 + k^2} = \sqrt{(1/2)^2 + k^2} = \sqrt{1/4 + k^2}$.The circle $S_2$ touches the circle $x^2 + y^2 = 9$ (which has center $(0, 0)$ and radius $R = 3$).For internal contact,the distance between centers $d = R - r$.The center of $S_2$ is $(1/2, k)$ and the center of $S_1$ is $(0, 0)$.So,$d = \sqrt{(1/2)^2 + k^2} = r$.Thus,$r = 3 - r$ $\Rightarrow 2r = 3$ $\Rightarrow r = 3/2$.Now,$r^2 = 1/4 + k^2$ $\Rightarrow (3/2)^2 = 1/4 + k^2$ $\Rightarrow 9/4 = 1/4 + k^2$ $\Rightarrow k^2 = 8/4 = 2$.Therefore,$k = \pm \sqrt{2}$.The center is $(1/2, \pm \sqrt{2})$.

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

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