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(D) Let the required circle be $S: x^2+y^2+2gx+2fy+c=0$. Since it is orthogonal to $S_1: x^2+y^2+4x+3=0$ and $S_2: x^2+y^2-12x+35=0$,the centre $(h, k

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$\sqrt{15}$

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(D) Let the required circle be $S: x^2+y^2+2gx+2fy+c=0$. Since it is orthogonal to $S_1: x^2+y^2+4x+3=0$ and $S_2: x^2+y^2-12x+35=0$,the centre $(h, k)$ of the circle must lie on the radical axis of $S_1$ and $S_2$.Radical axis is $S_1 - S_2 = 0$,which gives $(4x+3) - (-12x+35) = 0$,so $16x - 32 = 0$,or $x = 2$.Thus,the centre of the required circle is $(2, k)$.Since the circle is orthogonal to $S_1$,the radius $r$ satisfies $r^2 = S_1(2, k) = 2^2 + k^2 + 4(2) + 3 = 4 + k^2 + 8 + 3 = k^2 + 15$.The radius is minimized when $k = 0$,giving $r^2 = 15$,so $r = \sqrt{15}$.Thus,the minimum radius is $\sqrt{15}$.

Find the minimum radius of the circle which is orthogonal to both the circles $x^2+y^2+4x+3=0$ and $x^2+y^2-12x+35=0$.

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