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

Example of Use of Equation of Circle in Complex Plane

The equation:

$z \paren {\overline z + 2} = 3$

is a quadratic equation with $2$ solutions:

$z = 1$

$z = -3$

Proof

\(\ds z \paren {\overline z + 2}\)

\(=\)

\(\ds 3\)

\(\ds \leadsto \ \ \)

\(\ds \size z^2 + 2 z\)

\(=\)

\(\ds 3\)

Modulus in Terms of Conjugate

\(\ds \leadsto \ \ \)

\(\ds z = \dfrac {3 - \size z^2} 2\)

\(\in\)

\(\ds \R\)

\(\ds \leadsto \ \ \)

\(\ds z^2 + 2 z - 3\)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds z\)

\(=\)

\(\ds 1 \text{ or } -3\)

$\blacksquare$

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: $71 \ \text {(d)}$