Circle — General Complex Equation

General second-degree equation in z and z̄ representing a circle. Centre = −a, radius = √(|a|²−b). Real circle requires |a|² > b. This is the complex analogue of x²+y²+2gx+2fy+c = 0, with a = g+if and b = c.

The standard form $|z - z_0| = r$ is geometrically clean but algebraically inconvenient. Expanding it gives the general complex equation of a circle — the form used when the centre and radius are not immediately obvious.

Expanding the Standard Form

Start from $|z - z_0|^2 = r^2$ and use $|w|^2 = w\bar{w}$:

$$(z - z_0)\overline{(z - z_0)} = r^2$$ $$(z - z_0)(\bar{z} - \bar{z}_0) = r^2$$

Expanding:

$$z\bar{z} - z_0 \bar{z} - \bar{z}_0 z + z_0\bar{z}_0 = r^2$$ $$z\bar{z} - z_0 \bar{z} - \bar{z}_0 z + |z_0|^2 - r^2 = 0$$

Identifying the General Form

Set $a = -z_0$ (so $z_0 = -a$) and $b = |z_0|^2 - r^2$:

$$z\bar{z} + \bar{a}z + a\bar{z} + b = 0, \qquad a \in \mathbb{C},\; b \in \mathbb{R}$$

This is the general second-degree equation in $z$ and $\bar{z}$ representing a circle.

Reading Off Centre and Radius

From $a = -z_0$: the centre is $-a$.

From $b = |z_0|^2 - r^2 = |a|^2 - r^2$: the radius is $r = \sqrt{|a|^2 - b}$.

Conditions:

• $|a|^2 > b$: real circle
• $|a|^2 = b$: point circle (radius zero)
• $|a|^2 < b$: imaginary (no real points)

Connection to Cartesian Form

Writing $a = g + if$ (real $g$, $f$) and $z = x + iy$:

$$z\bar{z} = x^2 + y^2, \quad \bar{a}z + a\bar{z} = 2\operatorname{Re}(\bar{a}z) = 2(gx + fy)$$

So the equation becomes:

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

This is exactly the Cartesian general circle with $c = b$. The complex form $a = g + if$ packages the two linear coefficients into a single complex number.