Loci

A locus (plural: loci) is the set of all points that satisfy a given condition. In coordinate geometry, loci produce equations of curves. This lesson covers the most common loci at A-Level and how to derive their equations, as required by the Edexcel 9MA0 specification.

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What Is a Locus?

A locus is a path traced out by a point that moves according to a specific rule or condition.

Examples of loci:

• The set of points equidistant from a fixed point → a circle.
• The set of points equidistant from two fixed points → the perpendicular bisector of the line segment joining them.
• The set of points equidistant from a fixed point and a fixed line → a parabola.

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Locus 1: Equidistant from a Fixed Point (Circle)

The set of all points at distance r from a fixed point C(a, b) is a circle:

(x − a)² + (y − b)² = r²

Example: Find the locus of points that are 4 units from (2, −3).

(x − 2)² + (y + 3)² = 16.

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Locus 2: Equidistant from Two Points (Perpendicular Bisector)