Overview

Treat $z=a+bi$ as the point $(a,b)$. Geometry problems become distance and locus questions.

Key Ideas

• $|z| = \sqrt{a^2+b^2}$ is the distance to the origin.
• $|z-w|$ is the distance between points $z$ and $w$.
• Equations like $|z-z_0|=r$ represent circles.

Core Skills

Translate to Geometry

Rewrite complex equations as geometric loci in the plane.

Use Distance Properties

Interpret $|z-w|$ as a distance, then apply geometry or coordinate formulas.

Recognize Standard Loci

Equations of the form $|z-a|+|z-b|=k$ describe ellipses, and $|z-a|-|z-b|=k$ describe hyperbolas.

Worked Example

Describe the set of all $z$ such that $|z-2i|=3$.

This is the set of points at distance $3$ from $(0,2)$, so it is a circle of radius $3$ centered at $(0,2)$.

More Examples

Example 1: Distance Between Points

Find $|3+4i - (1-2i)|$.

This is the distance between $(3,4)$ and $(1,-2)$, so it is $\sqrt{(2)^2+(6)^2} =\sqrt{40}=2\sqrt{10}$.

Example 2: Circle Center

Describe $|z-(2-3i)|=5$.

Circle centered at $(2,-3)$ with radius $5$.

Example 3: Locus of Midpoints

If $|z-1|=|z+1|$, describe the locus of $z$.

Points equidistant from $-1$ and $1$ lie on the imaginary axis.

Strategy Checklist

• Convert to coordinates $(a,b)$.
• Interpret absolute values as distances.
• Identify the geometric locus.

Common Pitfalls

• Interpreting $|z|$ as the absolute value of the real part.
• Forgetting that $|z-w|$ is a distance formula.
• Mixing up the center and radius in $|z-z_0|=r$.

Practice Problems