Circle of Apollonius in Complex Plane/Examples/mod z-3 over z+3 less than 2
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Example of Circle of Apollonius in Complex Plane
The inequality:
- $\cmod {\dfrac {z - 3} {z + 3} } < 2$
describes the exterior of a circle embedded in the complex plane whose radius is $4$ and whose center is $\paren {-5, 0}$.
Proof
\(\ds \cmod {\dfrac {z - 3} {z + 3} }\) | \(<\) | \(\ds 2\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \cmod {z - 3}\) | \(<\) | \(\ds 2 \cmod {z + 3}\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \cmod {x + i y - 3}\) | \(<\) | \(\ds 2 \cmod {x + i y + 3}\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \sqrt {\paren {x - 3}^2 + y^2}\) | \(<\) | \(\ds 2 \sqrt {\paren {x + 3}^2 + y^2}\) | Definition of Complex Modulus | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \paren {x - 3}^2 + y^2\) | \(<\) | \(\ds 4 \paren {\paren {x + 3}^2 + y^2}\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x^2 - 6 x + 9 + y^2\) | \(<\) | \(\ds 4 x^2 + 24 x + 36 + 4 y^2\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds 3 x^2 + 30 x + 27 + 3 y^2\) | \(>\) | \(\ds 0\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x^2 + 10 x + 9 + y^2\) | \(>\) | \(\ds 0\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \paren {x + 5}^2 - 25 + 9 + y^2\) | \(=\) | \(\ds 0\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \paren {x + 5}^2 + y^2\) | \(>\) | \(\ds 16\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \cmod {z + 5}\) | \(>\) | \(\ds 4\) |
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: $48 \ \text{(b)}$