| Equation | Locus | Key Parameters |
|---|---|---|
| |z - z0| = r | Circle | Center z0, radius r |
| |z - z0| < r | Open disk (interior) | Center z0, radius r |
| |z - z1| = |z - z2| | Perpendicular bisector | Of segment z1z2 |
| |z - z1|/|z - z2| = k, k!=1 | Apollonius circle | Related to z1, z2, k |
| arg(z - z0) = theta | Ray | From z0 at angle theta |
| arg) = theta | Arc of circle | Through z1, z2 |
| |z-z1| + |z-z2| = 2a | Ellipse | Foci z1, z2 |
| |z-z1| - |z-z2| = 2a | Hyperbola branch | Foci z1, z2 |
| Re(z) = c | Vertical line x=c | -- |
| Im(z) = c | Horizontal line y=c | -- |
Max/Min |z| for |z-z0| = r:
- Maximum: |z0| + r (farthest from origin)
- Minimum: max(0, |z0| - r) (nearest to origin; 0 if origin is inside circle)
Key strategy: When given a complex equation involving |z-...|, immediately identify it as a standard locus and use geometric reasoning rather than algebraic expansion.