Circle of Apollonius in Complex Plane/Examples/mod z-3 over z+3 = 2/Proof 2
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Example of Circle of Apollonius in Complex Plane
The equation:
- $\cmod {\dfrac {z - 3} {z + 3} } = 2$
describes a circle embedded in the complex plane whose radius is $4$ and whose center is $\paren {-5, 0}$.
Proof
| \(\ds \cmod {\dfrac {z - 3} {z + 3} }\) | \(=\) | \(\ds 2\) | ||||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \paren {\dfrac {z - 3} {z + 3} } \paren {\dfrac {\overline z - 3} {\overline z + 3} }\) | \(=\) | \(\ds 4\) | |||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds z \overline z + 5 \overline z + 5 z + 9\) | \(=\) | \(\ds 0\) | |||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \paren {z + 5} \paren {\overline z + 5}\) | \(=\) | \(\ds 16\) | |||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \cmod {z + 5}\) | \(=\) | \(\ds 4\) |
Hence the result by Equation of Circle in Complex Plane.
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Miscellaneous Problems: $48 \ \text{(a)}$