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(C) The general equation of a circle passing through the origin $(0, 0)$ is given by $x^2 + y^2 + 2gx + 2fy = 0$ ... $(i)$.Since the circle passes thr

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

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(C) The general equation of a circle passing through the origin $(0, 0)$ is given by $x^2 + y^2 + 2gx + 2fy = 0$ ... $(i)$.Since the circle passes through $(0, b)$,we substitute $x = 0$ and $y = b$ into $(i)$:$0^2 + b^2 + 2g(0) + 2f(b) = 0$ $\Rightarrow b^2 + 2fb = 0$ $\Rightarrow f = -\frac{b}{2}$.Since the circle passes through $(a, b)$,we substitute $x = a$ and $y = b$ into $(i)$:$a^2 + b^2 + 2g(a) + 2f(b) = 0$.Substituting $f = -\frac{b}{2}$ into this equation:$a^2 + b^2 + 2ag + 2(-\frac{b}{2})(b) = 0$ $\Rightarrow a^2 + b^2 + 2ag - b^2 = 0$ $\Rightarrow a^2 + 2ag = 0$ $\Rightarrow g = -\frac{a}{2}$.Substituting $g = -\frac{a}{2}$ and $f = -\frac{b}{2}$ back into $(i)$:$x^2 + y^2 + 2(-\frac{a}{2})x + 2(-\frac{b}{2})y = 0 \Rightarrow x^2 + y^2 - ax - by = 0$.

The equation of the circle passing through the points $(0, 0)$,$(0, b)$,and $(a, b)$ is

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