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.