What is a locus?
A locus is the set of all points that satisfy a given condition. To find its equation, let the moving point be P(x, y), translate the condition into an equation using distances, then simplify.
Key formula
Distance between P(x, y) and A(a, b):
PA = √[(x − a)² + (y − b)²]
Locus equidistant from A and B: PA = PB ⇒ PA² = PB²
Common loci
- Fixed distance r from a point C(a, b): a circle (x − a)² + (y − b)² = r².
- Equidistant from two points: the perpendicular bisector of the segment.
The method is always the same: represent the moving point by P(x, y), write every stated distance using the distance formula, form an equation from the given condition, then expand and simplify. Squaring both sides early removes the awkward square roots and usually leaves a neat linear or quadratic equation. When the condition fixes the distance from a single point, the x² and y² terms survive and the locus is a circle; when it compares distances from two points, those squared terms cancel and the locus is a straight line — the perpendicular bisector. Recognising which case you are in helps you predict the form of the final answer before you begin.
Worked example
Find the locus of P(x, y) equidistant from A(1, 2) and B(5, 4). Set PA² = PB²: (x − 1)² + (y − 2)² = (x − 5)² + (y − 4)². Expanding: x² − 2x + 1 + y² − 4y + 4 = x² − 10x + 25 + y² − 8y + 16. Simplifying: −2x − 4y + 5 = −10x − 8y + 41, giving 8x + 4y − 36 = 0, or 2x + y − 9 = 0.
Remember
- Always start by letting the moving point be P(x, y).
- Square the distances to remove the square roots.
- A "fixed distance from a point" locus is a circle.