Locus in the Complex Plane

Locus in the Complex Plane

## Definition / Concept

A locus is the set of all points satisfying a given condition involving a complex variable.

## Formula

For:

$$|z-z_0|=r$$

the locus is a circle with centre $z_0$ and radius $$r$$.

For:

$$|z-z_1|=|z-z_2|$$

the locus is the perpendicular bisector of the line joining $z_1$ and $z_2$.

For:

$$|z-z_1|+|z-z_2|=k$$

the locus is an ellipse when:

$$k>|z_1-z_2|$$

## Usage

- Used to identify geometric loci directly from complex-number conditions.