General Equation

General second-degree equation representing a circle. Coefficients of x² and y² must be equal and non-zero; no xy term.

Expand the standard form $(x - h)^2 + (y - k)^2 = r^2$:

$$x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = r^2$$

Rearranging:

$$x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0$$

Setting $g = -h$, $f = -k$, $c = h^2 + k^2 - r^2$:

$$x^2 + y^2 + 2gx + 2fy + c = 0$$

Conditions for a valid circle: A general second-degree equation $ax^2 + bxy + cy^2 + dx + ey + f = 0$ represents a circle if and only if:

1. Coefficient of $x^2$ = coefficient of $y^2$ (both equal, non-zero)
2. The $xy$ coefficient is zero (no rotation)
3. $g^2 + f^2 - c > 0$ (non-imaginary radius)