Loci & Coordinate Geometry Proofs

This lesson covers how to use coordinate geometry to establish geometric results and work with loci (sets of points satisfying geometric conditions). These problems combine algebraic skill with geometric insight and appear at the higher end of A-Level examinations. They often require you to set up coordinates, translate geometric conditions into equations, and prove results algebraically.

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What Is a Locus?

A locus (plural: loci) is the set of all points satisfying a given geometric condition.

Common examples:

• The locus of points equidistant from a fixed point is a circle.
• The locus of points equidistant from two fixed points is the perpendicular bisector of the segment joining them.
• The locus of points at a fixed distance from a line is a pair of parallel lines.

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Finding the Equation of a Locus

Method:

1. Let the variable point be P(x, y).
2. Translate the geometric condition into an algebraic equation involving x and y.
3. Simplify to obtain the equation of the locus.

Example 1: Find the locus of points equidistant from A(1, 0) and B(5, 0).

Let P(x, y) be equidistant from A and B.