Equation of Circle in Complex Plane/Examples/Radius 4, Center (0, 0)
Jump to navigation
Jump to search
Example of Use of Equation of Circle in Complex Plane
The equation:
- $z \overline z = 16$
describes a circle embedded in the complex plane whose center is at $\tuple {0, 0}$ and whose radius is $4$.
Proof
\(\ds z \overline z\) | \(=\) | \(\ds 16\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {x + i y} \paren {x - i y}\) | \(=\) | \(\ds 16\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x^2 + y^2\) | \(=\) | \(\ds 16\) |
The result follows from Equation of Circle in Cartesian Plane.
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Conjugate Coordinates: $116 \ \text{(a)}$