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:

$\cmod {z - \alpha} = r$

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

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

Letting $z = x + i y$ gives:

\(\ds \cmod {z + 2 - i}\)

\(=\)

\(\ds 4\)

\(\ds \sqrt {\paren {x + 2}^2 + \paren {y - 1}^2}\)

\(=\)

\(\ds 4\)

Definition of Complex Modulus

\(\ds \paren {x + 2}^2 + \paren {y - 1}^2\)

\(=\)

\(\ds 16\)

squaring both sides

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $13 \ \text{(a)}$