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:
it does not, because $y$ is real.
Otherwise, the circle would have center $\tuple {-1, -i} \notin \R^2$.
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: $71 \ \text {(d)}$