Equation of Circle in Complex Plane/Examples/Imaginary Radius 2, Center (2, 0)

Example of Use of Equation of Circle in Complex Plane

The equation:

$z \overline z - 2 z - 2 \overline z + 8 = 0$

describes a circle embedded in the complex plane whose center is at $\tuple {2, 0}$ and whose radius is $2$ imaginary units.

$\ds z \overline z - 2 z - 2 \overline z + 8$

$=$

$\ds 0$

$\ds \leadsto \ \ $

$\ds \paren {x + i y} \paren {x - i y} - 2 \paren {x + i y} - 2 \paren {x - i y} + 8$

$=$

$\ds 0$

$\ds \leadsto \ \ $

$\ds x^2 + y^2 - 4 x - 2 i y + 2 i y + 8$

$=$

$\ds 0$

$\ds \leadsto \ \ $

$\ds \paren {x - 2}^2 - 4 + y^2 + 8$

$=$

$\ds 0$

$\ds \leadsto \ \ $

$\ds \paren {x - 2}^2 + y^2$

$=$

$\ds -4$

The result follows from Equation of Circle in Cartesian Plane, except that the square of the radius is $-4$.

Hence the radius has do be described as $2$ imaginary units.

$\blacksquare$

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