Formulas/maths/Circles/Radius from General Form

Radius from General Form

Radius of x² + y² + 2gx + 2fy + c = 0. Real circle requires g² + f² − c > 0; point circle if = 0; imaginary if < 0.

Derivation

From the completed-square form derived in the centre derivation:

(x + g)^2 + (y + f)^2 = g^2 + f^2 - c

The right side equals $r^2$ , so:

r = \sqrt{g^2 + f^2 - c}

Three cases:

Condition	Geometric object
$g^2 + f^2 - c > 0$	Real circle with centre $(-g, -f)$
$g^2 + f^2 - c = 0$	Point circle — radius zero, just the centre
$g^2 + f^2 - c < 0$	Imaginary circle — no real points

In JEE problems, a given general equation may need this check before proceeding.