If the equation ax^2 + by^2 + 2hxy + 2gx + 2f | vedclass

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(C) The general equation of a second-degree curve is $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$. For this to represent a circle,the coefficient of $x^2$

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$a = b, h = c = 0$

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(C) The general equation of a second-degree curve is $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$. For this to represent a circle,the coefficient of $x^2$ must equal the coefficient of $y^2$ (i.e.,$a = b$) and the coefficient of $xy$ must be zero (i.e.,$h = 0$). Thus,the equation becomes $a(x^2 + y^2) + 2gx + 2fy + c = 0$. Since the circle passes through the origin $(0, 0)$,substituting $x = 0$ and $y = 0$ into the equation gives $a(0)^2 + a(0)^2 + 2g(0) + 2f(0) + c = 0$,which implies $c = 0$. Therefore,the conditions are $a = b, h = 0, c = 0$.

If the equation $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$ represents a circle passing through the origin,then

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