Let the equation ax^2+2hxy+by^2+2gx+2fy+c=0 r | vedclass

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(A) Given equation: $ax^2+2hxy+by^2+2gx+2fy+c=0$ $(i)$ If equation $(i)$ represents a circle,then $a=b$ and $h=0$. Substituting these in $(i)$,we get:

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$bc > 0$

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(A) Given equation: $ax^2+2hxy+by^2+2gx+2fy+c=0$ $(i)$ If equation $(i)$ represents a circle,then $a=b$ and $h=0$. Substituting these in $(i)$,we get: $bx^2+by^2+2gx+2fy+c=0$. Dividing by $b$ (since $b 
eq 0$): $x^2+y^2+2(\frac{g}{b})x+2(\frac{f}{b})y+\frac{c}{b}=0$. For a point circle,the radius $r$ must be $0$. The radius of a circle $x^2+y^2+2gx+2fy+c=0$ is $\sqrt{g^2+f^2-c}$. Here,$g' = \frac{g}{b}$,$f' = \frac{f}{b}$,and $c' = \frac{c}{b}$. So,$r^2 = (\frac{g}{b})^2 + (\frac{f}{b})^2 - \frac{c}{b} = 0$. This implies $\frac{g^2+f^2}{b^2} = \frac{c}{b}$. Multiplying by $b^2$,we get $g^2+f^2 = bc$. Since $g^2+f^2 \geq 0$,it follows that $bc \geq 0$. For a point circle other than the origin,$g^2+f^2 > 0$,therefore $bc > 0$.

Let the equation $ax^2+2hxy+by^2+2gx+2fy+c=0$ represent a point circle other than the origin. Then which one of the following conditions must hold?

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