Distance-Based Loci

Distance-Based Loci

## Formula

$$|z-z_1|=k$$

represents a circle with centre $z_1$ and radius $$k$$.

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

represents the perpendicular bisector of the segment joining $z_1$ and $z_2$.

$$|z-z_1|<k$$

represents the interior of a circle.

$$|z-z_1|>k$$

represents the exterior of a circle.

## Conditions / Special Cases

For:

$$|z-z_1|\le k$$

the locus is the closed circular region.

For:

$$|z-z_1|\ge k$$

the locus is the exterior including the boundary.

## Usage

- Used to interpret inequalities involving complex moduli geometrically.