Equation of Circle in Complex Plane/Examples/z (conj z + 2) = 3/Mistake

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Source Work

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.)

Chapter $1$: Complex Numbers
Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $71 \ \text {(d)}$


Mistake

Describe and graph the locus represented by each of the following:
... $\text (d)$ $z \paren {\overline z + 2} = 3$
Ans. ... $\text (d)$ circle, ...


Correction

Working through in the direction one would go when trying to demonstrate the locus is a circle:

\(\ds z \paren {\overline z + 2}\) \(=\) \(\ds 3\)
\(\ds \leadsto \ \ \) \(\ds z \overline z + 2 z\) \(=\) \(\ds 3\)
\(\ds \leadsto \ \ \) \(\ds \paren {x + i y} \paren {x - i y} + 2 x + 2 i y\) \(=\) \(\ds 3\)
\(\ds \leadsto \ \ \) \(\ds x^2 + y^2 + 2 x + 2 i y\) \(=\) \(\ds 3\)
\(\ds \leadsto \ \ \) \(\ds \paren {x + 1}^2 - 1 + \paren {y + i}^2 - i^2\) \(=\) \(\ds 3\)
\(\ds \leadsto \ \ \) \(\ds \paren {x + 1}^2 + \paren {y + i}^2\) \(=\) \(\ds 3\)

While it looks like the result follows from Equation of Circle in Cartesian Plane:

Equation of Circle in Complex Plane-Example-z (conj z + 2) = 3.png

it does not, because $y$ is real.

Otherwise, the circle would have center $\tuple {-1, -i} \notin \R^2$.


Sources