Circle of Apollonius in Complex Plane/Examples/mod z-3 over z+3 = 2/Proof 1

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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

A point $P$ on this circle is $2$ times the distance from $z = 3$ as it is from $z = -3$.

Circle-of-Apollonius-(z-3)-(z+3).png

Let $z = x + i y$.

\(\ds \cmod {\dfrac {z - 3} {z + 3} }\) \(=\) \(\ds 2\)
\(\ds \leadstoandfrom \ \ \) \(\ds \cmod {z - 3}\) \(=\) \(\ds 2 \cmod {z + 3}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \cmod {x + i y - 3}\) \(=\) \(\ds 2 \cmod {x + i y + 3}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \sqrt {\paren {x - 3}^2 + y^2}\) \(=\) \(\ds 2 \sqrt {\paren {x + 3}^2 + y^2}\) 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.

$\blacksquare$


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