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(C) The general equation of a circle passing through the origin is $x^2 + y^2 + 2gx + 2fy = 0$.For a system of co-axial circles,the radical axis of an

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$x^2 + y^2 - 2ax = 0$

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(C) The general equation of a circle passing through the origin is $x^2 + y^2 + 2gx + 2fy = 0$.For a system of co-axial circles,the radical axis of any two circles must be the same.The given circles are $S_1: x^2 + y^2 - a^2 = 0$ and $S_2: x^2 + y^2 + 2ax - 2a^2 = 0$.The radical axis of $S_1$ and $S_2$ is $S_1 - S_2 = 0$,which gives $2ax - a^2 = 0$,or $x = \frac{a}{2}$.Let the required circle be $S_3: x^2 + y^2 + 2gx + 2fy = 0$.The radical axis of $S_1$ and $S_3$ is $x^2 + y^2 - a^2 - (x^2 + y^2 + 2gx + 2fy) = 0$,which simplifies to $2gx + 2fy + a^2 = 0$.Since the circles are co-axial,this radical axis must be $x = \frac{a}{2}$.Comparing $2gx + 2fy + a^2 = 0$ with $x = \frac{a}{2}$ (or $x - \frac{a}{2} = 0$),we get $f = 0$ and $\frac{2g}{1} = \frac{a^2}{-a/2} = -2a$.Thus,$g = -a$.Substituting $g = -a$ and $f = 0$ into the equation of $S_3$,we get $x^2 + y^2 - 2ax = 0$.

The equation of a circle passing through the origin and co-axial to the circles $x^2 + y^2 = a^2$ and $x^2 + y^2 + 2ax = 2a^2$ is

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