Geometric sets defined by a condition — a single word that unifies circles, perpendicular bisectors, and parallel lines.
The Mnemonic
One Word for an Entire Category of Shapes
A locus (plural: loci) is the set of ALL points that satisfy one specific given condition. This single concept unifies several shapes already covered in this course, once you notice they're really just answers to different "what set of points satisfies THIS condition" questions.
A circle IS the locus of all points a fixed distance (the radius) from a center point. A perpendicular bisector IS the locus of all points equidistant from two given endpoints. Even a straight line can be described as a locus: the set of all points satisfying a given linear equation.
A circle is a locus: the set of ALL points satisfying one condition (fixed distance from center)
💡 Memory Trick
"Geometric sets defined by a condition." Whenever you see the phrase "the set of all points such that...", that's a locus problem. The whole task is translating a plain-English condition into an equation or shape that captures EVERY point satisfying it — not just one example point.
Why It Works
Reframing Familiar Shapes as Loci
This lesson isn't introducing new formulas — it's showing that several formulas you already know were secretly answering locus questions the whole time. The Equation of a Circle, (x−h)²+(y−k)²=r², answers "what is the locus of points exactly r from (h,k)?" The Perpendicular Bisector equation answers "what is the locus of points equidistant from A and B?"
Recognizing a problem as a locus problem is valuable specifically because it tells you WHICH earlier tool to reach for — a distance-based condition points to the Distance Formula or Circle Equation; an equidistant-from-two-points condition points to the Perpendicular Bisector; an equidistant-from-two-lines condition (a related idea) points to an angle bisector.
Using It In A Proof
Translating a Condition Into an Equation
The core skill in a locus problem is reading the plain-English description and recognizing which earlier geometric tool actually answers it.
1
Identify the type of condition
Fixed distance from ONE point → circle. Equal distance from TWO points → perpendicular bisector. A specific linear relationship → a line equation.
2
Match to the correct formula
Once the condition type is identified, the actual equation-building uses tools already covered — Distance Formula, Circle Equation, Perpendicular Bisector, etc.
3
State the locus as a complete geometric description
The final answer should describe the actual shape (a circle of radius X centered at Y, or a line with a specific equation) — not just a single example point that happens to satisfy the condition.
Full Worked Example
Finding the Locus of Points a Fixed Distance From a Point
Given: Find the locus of all points exactly 5 units from the point (3, −2).
1
Identify the condition type
"Exactly 5 units from a point" is a fixed-distance-from-one-point condition — this is a circle.
2
Identify center and radius
Center = (3, −2), radius = 5.
3
Write the equation using the Circle Equation formula
(x−3)² + (y−(−2))² = 5², which simplifies to (x−3)² + (y+2)² = 25.
4
State the complete locus description
The locus is a circle centered at (3, −2) with radius 5, described by (x−3)² + (y+2)² = 25.
This is exactly the Equation of a Circle lesson, reframed — recognizing the locus TYPE (fixed distance from one point) is what told us which earlier formula to reach for.
🎯 Quick Worked Example
Find the locus of all points equidistant from A(1, 2) and B(7, 2).
1
Identify the condition type. "Equidistant from two points" is exactly the perpendicular bisector condition.
2
Find the midpoint. M = ((1+7)/2, (2+2)/2) = (4, 2).
3
Find the perpendicular slope and describe the locus. Slope of AB = (2−2)/(7−1) = 0 (horizontal), so the perpendicular bisector is vertical: x = 4. The locus is the vertical line x = 4.
📌 Exam Application
Locus problems are frequently phrased in plain English without any hint of which formula to use ("find all points 4 units from...", "find all points equally far from...") — the real test is translating the WORDS into the correct geometric category (circle vs. perpendicular bisector vs. line) before any calculation begins.
⚠️ Most Common Locus Problems Mistakes
Trap 1 — Giving one example point instead of describing the whole locus: A locus is an entire set of infinitely many points, not a single point — the final answer needs to be a full equation or shape description, not just one coordinate pair that happens to work.
Trap 2 — Misreading 'equidistant from two points' as 'equidistant from two lines': These are genuinely different loci — equidistant from two POINTS gives a perpendicular bisector; equidistant from two LINES gives an angle bisector — always check exactly what the condition is measuring distance to.
✓ Quick Self-Test
1) What is a locus? 2) What locus is described by 'all points a fixed distance from a center'? 3) What locus is described by 'all points equidistant from two given points'? 4) Find the locus of all points 7 units from the point (0, 0). 5) Find the locus of all points equidistant from (2, 5) and (2, 9).