Loci and Locus Regions
A locus is the set of all points that satisfy a given geometric condition, and a locus region is the area of the plane in which an inequality-type condition holds. In brief: the locus of points at a fixed distance r from a point O is a circle with radius r, the locus of points equally far from two points is their perpendicular bisector, and the locus of points equally far from two intersecting lines is the pair of angle bisectors.
Loci turn verbal conditions such as “all points at most 3 km from the transmitter” into precise shapes that can be drawn and calculated. Many problems combine several conditions, and the answer is the intersection of the individual loci. This guide lists the standard loci, shows how to combine conditions, how to describe regions and how to use coordinates to write equations of loci.
What are the standard loci?
These five loci are the building blocks.
- Fixed distance from a point: a circle with the point as center.
- Fixed distance from a line: a pair of parallel lines, one on each side, at the given distance.
- Equidistant from two points A and B: the perpendicular bisector of AB.
- Equidistant from two intersecting lines: the two angle bisectors (a pair of perpendicular lines).
- Equidistant from two parallel lines: the parallel line halfway between them.
A segment of fixed length with ends on two perpendicular lines has a midpoint that moves on a circle: the locus of the midpoint of a sliding ladder is a quarter circle with the corner as center.
What are locus regions?
When the condition is an inequality, the locus is a region.
- “Less than 4 cm from point O”: the inside of a circle of radius 4 cm (the boundary is excluded).
- “At least 4 cm from O”: the outside of that circle, including the circle.
- “Closer to A than to B”: the half-plane on A’s side of the perpendicular bisector of AB.
- “Within 2 cm of a line segment AB”: a stadium-shaped region: a rectangle of width 4 cm along AB with a half-disc at each end.
- “Closer to line l1 than to l2“: the region between the angle bisectors on the side of l1, i.e., one of the two pairs of opposite angular regions.
Draw the boundary as a solid line if it is included (≤ or ≥) and as a dashed line if not (< or >). Shade the region.
How do you combine several conditions?
- Draw the figure to scale.
- Draw the locus for each condition separately.
- Shade or mark the points that satisfy all conditions: the intersection of the loci.
- Describe the result in words and, if asked, measure or calculate it.
Example 1: a treasure
A treasure is buried at a point that is exactly 5 m from a tree T and equally far from two rocks A and B, which are 8 m apart, with T on the perpendicular bisector of AB. The locus of points 5 m from T is a circle. The locus of points equally far from A and B is the perpendicular bisector. The line through T meets the circle in two points, 5 m before and beyond T. So there are two possible places.
Example 2: a goat on a rope
A goat is tied with a 3 m rope to a corner of a square shed with sides 4 m. The shed blocks one quarter of the plane around the corner, and the rope is shorter than the side 4 m, so it cannot wrap around the shed. The region the goat can reach is therefore a three-quarter disc of radius 3 m. Its area is (3/4) · π · 32 = 6.75π ≈ 21.2 m2.
Example 3: a mobile network
A radio mast covers points within 10 km. A second mast, 16 km away, also covers 10 km. The overlap region is where both signals are available. The two circles meet at two points; the distance from each mast to the common chord is 8 km, so the half-length of the chord is √(100 − 64) = 6 km, and the chord has length 12 km.
How do you find the equation of a locus?
Let P(x, y) be a general point and write the condition with the distance formula, then simplify.
Example 4: equidistant from two points
A(0, 0) and B(6, 0). PA = PB means PA2 = PB2: x2 + y2 = (x − 6)2 + y2, so 0 = −12x + 36 and x = 3. This is the perpendicular bisector.
Example 5: at a distance of 2 from a point
Center (1, −1): (x − 1)2 + (y + 1)2 = 4. This is a circle with radius 2.
Example 6: twice as far from A as from B
Let A(0, 0) and B(3, 0), PA = 2PB. Then x2 + y2 = 4[(x − 3)2 + y2], so 3x2 − 24x + 3y2 + 36 = 0, or x2 − 8x + y2 + 12 = 0. Completing the square: (x − 4)2 + y2 = 4. The locus is a circle with center (4, 0) and radius 2 (a circle of Apollonius).
Table of standard loci
| Condition | Locus | Example equation |
|---|---|---|
| Distance 3 from O(0, 0) | circle | x2 + y2 = 9 |
| Distance 2 from the x-axis | two parallel lines | y = 2 and y = −2 |
| Equidistant from (0, 0) and (6, 0) | perpendicular bisector | x = 3 |
| Equidistant from the axes | two angle bisectors | y = x and y = −x |
| Equidistant from y = 1 and y = 5 | midline | y = 3 |
| Within 3 of the origin | disc (region) | x2 + y2 < 9 |
Where does the idea come from?
The word locus is Latin for “place”. Greek geometers studied loci in depth: the circle is defined as a locus, and Apollonius wrote a book on plane loci. Pappus later discussed a famous problem of loci to three or four lines, which Descartes solved in the seventeenth century with coordinates, an achievement that helped to lay the foundations of analytic geometry.
Where are loci used today?
Coverage of radio and Wi-Fi signals, the range of a security camera and flight paths near an airport are loci. Distance-based regions underlie Voronoi diagrams, which divide a map into cells closest to each facility, and are used in planning the locations of hospitals and schools. Computer graphics use loci to define curves and shapes.
Which mistakes are common?
Giving one point instead of the whole set
A locus is a complete set of points. Do not stop after finding one solution.
Forgetting the second line
The points at a distance d from a line lie on two parallel lines, one on each side. Likewise, the points equidistant from two intersecting lines lie on two bisectors.
Mixing solid and dashed boundaries
Use a dashed line for strict inequalities.
Not checking by a test point
Pick a point in the region and verify that it meets all conditions.