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

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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:

\(\ds \cmod {z + 3 - 4 i}\) \(=\) \(\ds 2\)
\(\ds \sqrt {\paren {x + 3}^2 + \paren {y - 4}^2}\) \(=\) \(\ds 2\) Definition of Complex Modulus
\(\ds \paren {x + 3}^2 + \paren {y - 4}^2\) \(=\) \(\ds 4\) squaring both sides

$\blacksquare$


Sources

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