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:

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

Proof

From Equation of Circle in Complex Plane, a circle whose radius is $r$ and whose center is $\alpha$ has equation:

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

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

$\cmod {z - i} = 2$

Letting $z = x + i y$ gives:

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

\(=\)

\(\ds 2\)

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

\(=\)

\(\ds 2\)

Definition of Complex Modulus

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

\(=\)

\(\ds 4\)

squaring both sides

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $71 \ \text {(a)}$