$\begin{array}{1 1}(A)\;\bigg(\frac{1}{2},\frac{1}{2}\bigg) \\(B)\;\bigg(\frac{1}{2},-\sqrt 2\bigg) \\(C)\;\bigg(\frac{3}{2},\frac{1}{2}\bigg) \\(D)\;\bigg(\frac{1}{2},\frac{3}{2}\bigg) \end{array}$

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Let the requirement circle be

$x^2+y^2+2gx+2fy+c=0$

Since it passes through $(0,0)$ and $(1,0)$

=> $C=0$ and $g=\large\frac{-1}{2}$

Points $(0,0)$ and $(1,0)$ lie inside the circle $x^2+y^2=9$ So two circles touch internally

=> $C_1C_2= r_1-r_2$

$\sqrt {g^2+f^2}=3 -\sqrt {g^2+f^2}$

=> $ \sqrt {g^2+f^2}=\large\frac{3}{2}$

=> $ f^2=\large\frac{9}{4}-\frac{1}{4}$$=2$

Hence the centres of required circle are $\bigg(\large\frac{1}{2},$$-\sqrt 2\bigg)$

Hence B is the correct answer.

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