Equation of Circle in Complex Plane/Examples/Radius 4, Center (-2, 1)

Example of Use of Equation of Circle in Complex Plane
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:
 * $\paren {x + 2}^2 + \paren {y - 1}^2 = 16$

Proof
From Equation of Circle in Complex Plane, a circle whose radius is $r$ and whose center is $\alpha$ has equation:
 * $\cmod {z - \alpha} = r$


 * Equation of Circle in Complex Plane-Example-Radius 4, Center (-2, 1).png

Substituting $\alpha = -2 + i$ and $r = 4$ gives:


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

Letting $z = x + i y$ gives: