Equation of Circle in Complex Plane/Examples

Examples of Use of Equation of Circle in Complex Plane

Radius $4$, Center $\tuple {-2, 1}$

Let $C$ be a circle embedded in the complex plane whose radius is $4$ and whose center is $\paren {-2, 1}$.

Then $C$ can be described by the equation:

$\cmod {z + 2 - i} = 4$

or in conventional Cartesian coordinates:

$\paren {x + 2}^2 + \paren {y - 1}^2 = 16$

Radius $2$, Center $\tuple {0, 1}$

Let $C$ be a circle embedded in the complex plane whose radius is $2$ and whose center is $\paren {0, 1}$.

Then $C$ can be described by the equation:

$\cmod {z - i} = 2$

or in conventional Cartesian coordinates:

$x^2 + \paren {y - 1}^2 = 4$

Radius $2$, Center $\tuple {-3, 4}$

Let $C$ be a circle embedded in the complex plane whose radius is $2$ and whose center is $\paren {-3, 4}$.

Then $C$ can be described by the equation:

$\cmod {z + 3 - 4 i} = 2$

or in conventional Cartesian coordinates:

$\paren {x + 3}^2 + \paren {y - 4}^2 = 4$

Radius $4$, Center $\tuple {0, -3}$

The inequality:

$\cmod {z + 3 i} > 4$

describes the area outside the circle whose center is at $-3 i$, whose radius is $4$.

Annulus: $1 < \cmod {z + i} \le 2$

The inequality:

$1 < \cmod {z + i} \le 2$

describes the inside of the annulus whose center is at $-i$, whose inner radius is $1$ and whose outer radius is $2$.

This annulus does not include its inner boundary, but does include its outer boundary.

Circle Defined by $z \paren {\overline z + 2} = 3$

The equation:

$z \paren {\overline z + 2} = 3$

is a quadratic equation with $2$ solutions:

$z = 1$

$z = -3$

Radius $4$, Center $\tuple {0, 0}$

The equation:

$z \overline z = 16$

describes a circle embedded in the complex plane whose center is at $\tuple {0, 0}$ and whose radius is $4$.

Imaginary Radius $2$, Center $\tuple {2, 0}$

The equation:

$z \overline z - 2 z - 2 \overline z + 8 = 0$

describes a circle embedded in the complex plane whose center is at $\tuple {2, 0}$ and whose radius is $2$ imaginary units.

Degenerate Case: Straight Line $x = 2$

The equation:

$z + \overline z = 4$

describes the straight line $x = 2$ embedded in the complex plane

Degenerate Case: Straight Line $y = -3$

The equation:

$\overline z = z + 6 i$

describes the straight line $y = -3$ embedded in the complex plane