Equation of Circle in Complex Plane/Examples/Radius 2, Center (-3, 4)

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 $\paren {-3, 4}$.

Then $C$ can be described by the equation:
 * $\cmod {z + 3 - 4 i} = 2$

or in conventional Cartesian coordinates:
 * $\paren {x + 3}^2 + \paren {y - 4}^2 = 4$

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

Substituting $\alpha = -3 + 4 i$ and $r = 2$ gives:


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

that is:


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

Letting $z = x + i y$ gives: