Loci in Plane Geometry: Definitions and Key Examples

A locus (plural: loci) is the set of all points satisfying a given condition. Instead of one point, a locus describes a whole path or region: every point on it meets the condition, and every point meeting the condition lies on it.

Loci connect geometry with algebra: the locus idea turns "points at distance 5 from the origin" into the equation of a circle. They appear in coordinate geometry, navigation, and anywhere a moving point must obey a fixed rule.

Key Concepts

• Definition: a locus is a set of points satisfying geometric conditions — it can be a line, a curve, or a region.
• Fixed distance from a point: a circle centered at that point with the given distance as radius.
• Equidistant from two fixed points: the perpendicular bisector of the segment joining them.
• Equidistant from two intersecting lines: the pair of angle bisectors (two perpendicular lines).
• Fixed distance from a line: a pair of lines parallel to it, one on each side at that distance.

How It Works

• To find a locus, test points: pick several points satisfying the condition, plot them, and observe the pattern — then argue that every point of that pattern qualifies.
• Equidistant from A and B: any point P with PA = PB lies on the perpendicular bisector of AB (provable with congruent triangles).
• Distance from a point: fixing distance r from center O gives x² + y² = r² — the circle equation is a locus equation.
• Combining conditions: points satisfying two conditions lie at the intersection of the two loci.

Worked Example

Points A(2, 1) and B(8, 1) are fixed. Describe the locus of points P such that PA = PB, and give its equation.

1. The condition PA = PB means P is equidistant from A and B, so the locus is the perpendicular bisector of AB.
2. Midpoint of AB: ((2 + 8)/2, (1 + 1)/2) = (5, 1).
3. AB is horizontal, so its perpendicular bisector is the vertical line through x = 5.
4. The locus is the line x = 5. Check: P(5, 4) gives PA = √(9 + 9) = √18 and PB = √(9 + 9) = √18. Equal. Correct.

Common Misconceptions

• A locus is a set, not a single point. "The locus is the midpoint" is wrong when infinitely many points qualify — here the locus is the whole bisector line.
• Equidistant from two points gives a line. Only the midpoint is the closest such point, but every point on the perpendicular bisector is equidistant.

Real-World Applications

• GPS navigation: each satellite defines a circle (locus of constant distance); your position is where the loci intersect.
• Mobile networks: the boundary between two tower coverage zones is the perpendicular bisector locus of equal signal distance.
• Robotics: a robot arm joint traces a circular locus, and path planning keeps the arm on allowed loci while avoiding obstacles.
• Safety zones: "stay at least 50 m from the runway" defines a locus region used in airport planning.

A locus is the complete set of points satisfying a condition — a circle for fixed distance from a point, a perpendicular bisector for equal distance from two points, and angle bisectors for equal distance from two lines. Describe the set fully, verify it in both directions, and loci turn from abstract definitions into practical tools for coordinates, navigation, and design.