Equation for Perpendicular Bisector of Two Points in Complex Plane/Examples/2+i, 3-2i/Standard Form
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Example of Use of Equation for Perpendicular Bisector of Two Points in Complex Plane
Let $L$ be the perpendicular bisector of the straight line through $2 + i$ and $3 - 2 i$ in the complex plane.
Then $L$ can be expressed by the equation:
- $x - 3 y = 4$
Proof
Let $z_1 = 2 + i$ and $z_2 = 3 - 2 i$.
From Equation for Perpendicular Bisector of Two Points in Complex Plane/Standard Form, $L$ can be expressed by the equation:
- $\map \Re {z_2 - z_1} x + \map \Im {z_2 - z_1} y = \dfrac {\cmod {z_2}^2 - \cmod {z_1}^2} 2$
We have:
- $z_2 - z_1 = 1 - 3 i$
- $\cmod z_1^2 = 2^2 + 1^2 = 5$
- $\cmod z_2^2 = 3^2 + 2^2 = 13$
and hence we have the equation:
- $x - 3 y = \dfrac {13 - 5} 2 = 4$
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $70 \ \text {(b)}$