Equation of Circle in Complex Plane/Examples/Straight Line x = 2

Example of Use of Equation of Circle in Complex Plane

The equation:

$z + \overline z = 4$

describes the straight line $x = 2$ embedded in the complex plane

Proof

This is an instance of the Equation of Circle in Complex Plane: Formulation 2:

$\alpha z \overline z + \beta z + \overline \beta \overline z + \gamma = 0$

where $\alpha = 0$.

This is a straight line if and only if $\alpha = 0$ and $\beta \ne 0$.

Hence:

\(\ds z + \overline z\)

\(=\)

\(\ds 4\)

\(\ds \leadsto \ \ \)

\(\ds \paren {x + i y} + \paren {x - i y}\)

\(=\)

\(\ds 4\)

\(\ds \leadsto \ \ \)

\(\ds 2 x\)

\(=\)

\(\ds 4\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds 2\)

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Conjugate Coordinates: $116 \ \text{(c)}$