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

## Source Work

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 $\tuple {-1, -i} \notin \R^2$.