Prove that the equation $x^2+y^2+2gx+2fy+c=0$ always represents a circle.
I just don't have any idea regarding this. Can anyone help me?
Help much appreciated!
Thanks..
2026-04-24 06:44:42.1777013082
Equation of Circle
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\begin{align*} x^2 + y^2 +2gx + 2fy + c = 0 &\iff (x + g)^2 - g^2 + (y + f)^2 - f^2 + c = 0 \\ &\iff (x + g)^2 + (y + f)^2 = g^2 + f^2 - c \end{align*} which is the equation for a circle with center $(-g,-f)$ and radius $\sqrt{g^2 + f^2 -c}$. Note that we demand that $g^2 + f^2 > c$.