Complex loci on an Argand diagram present a fascinating intersection of algebra and geometry. This comprehensive guide aims to provide a deep understanding of complex loci, their properties, and their applications in solving complex mathematical problems.

Image courtesy of Third Space Learning

## Loci Basics

- Loci in complex numbers: Set of points meeting a specific condition.
- Shown on Argand diagram as complex numbers forming shapes or paths.
- Key to understanding complex number geometry.

## Loci Equations

- Example: $|z - a| = k$ (with '$z$' as a complex number, '$a$' as a fixed complex number, '$k$' as a positive real number).
- Represents a circle on Argand diagram, center '$a$', radius '$k$'.
- Important for visualizing complex number operations.

## Loci Inequalities

- Inequalities, not just equations, define loci.
- Example: |z - a| < k shows a region inside a circle, center '$a$', radius '$z$'.
- Crucial for solving problems with regions and boundaries in complex numbers.

## Examples

### Example 1. Equation: $|z - (3 + 4i)| = 5$

**Step 1:**$z = x + yi$, equation becomes $|(x + yi) - (3 + 4i)| = 5$.**Step 2:**Simplify to $|(x - 3) + (y - 4)i| = 5$.**Step 3:**Use modulus, $\sqrt{(x - 3)^2 + (y - 4)^2} = 5$.**Step 4:**Square both sides, $(x - 3)^2 + (y - 4)^2 = 25$.**Result:**Circle with center (3, 4) and radius 5.

### Example 2. Inequality: |z + 2 - 3i| < 4

**Step 1:**$z = x + yi$, inequality becomes |(x + yi) + 2 - 3i| < 4.**Step 2:**Simplify to |(x + 2) + (y - 3)i| < 4.**Step 3:**Use modulus, \sqrt{(x + 2)^2 + (y - 3)^2} < 4.**Result:**Interior of a circle, center (-2, 3), radius 4.

The graph shows these loci:

- The blue circle represents the first equation's locus, a circle with center at (3, 4) and radius 5.
- The green area represents the second inequality's locus, the interior of a circle centered at (-2, 3) with radius 4.

Written by: Dr Rahil Sachak-Patwa

LinkedIn

Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.