Equation of Circle in Complex Plane/Examples/Radius 2, Center (0, 1)

Example of Use of Equation of Circle in Complex Plane
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:
 * $\paren x^2 + \paren {y - 1}^2 = 4$

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 2, Center (0, 1).png

Substituting $\alpha = i$ and $r = 2$ gives:


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

Letting $z = x + i y$ gives: