A locus (plural: loci) is the set of all points that satisfy a given condition or rule.
| Condition | Locus |
|---|---|
| Fixed distance r from point O | Circle, centre O, radius r |
| Equidistant from two points A and B | Perpendicular bisector of AB |
| Equidistant from two lines | Angle bisector |
| Fixed distance from a line | Two parallel lines |
The locus of all points at distance r from a fixed point O is a circle with centre O and radius r.
A dog is tied to a post with a 5m rope. The locus of the dog's maximum reach is a circle of radius 5m centred on the post.
Describe the locus of a point P that moves so that |OP| = 3 cm, where O is origin.
Locus: Circle, centre O(0,0), radius 3 cm. Equation: xΒ² + yΒ² = 9
Points within distance r from O but at distance β₯ rβ β annulus (ring) between two circles.
The locus of all points equidistant from two fixed points A and B is the perpendicular bisector of AB.
A(2,0) and B(8,0). Find equation of locus of points equidistant from A and B.
Midpoint = (5,0). AB is horizontal, so perp. bisector is vertical: x = 5
A mobile phone tower is to be built equidistant from two towns. The ideal location lies on the perpendicular bisector of the line joining the two towns.
The locus of points equidistant from two intersecting lines is the angle bisector of the angle between the lines.
1. Mark equal distances along each line from intersection point.
2. Draw arcs from these points.
3. Join intersection of arcs to the vertex.
Finding the location of a fire station equidistant from two roads uses the angle bisector locus.
When a point must satisfy two conditions simultaneously, find the intersection of both loci.
Find the locus of points that are:
(i) 4 cm from point A, AND
(ii) equidistant from two points B and C
Draw circle radius 4cm centred at A. Draw perpendicular bisector of BC. The intersection points satisfy both conditions.
Construct a point P such that PA = 3cm and PB = 5cm where AB = 6cm.
Draw circle radius 3cm at A and circle radius 5cm at B. Intersection gives P.
| Condition | Locus/Region |
|---|---|
| PA = r | Circle, radius r |
| PA β€ r | Interior of circle (disk) |
| PA β₯ r | Exterior of circle |
| PA < PB | Region closer to A (one side of perp. bisector) |
Shade the region where a point is within 3cm of A AND closer to A than to B.
Draw circle radius 3 at A. Draw perpendicular bisector of AB. Shade region inside circle on A's side of perp. bisector.
A hospital must be built within 10km of Town A and within 8km of Town B. The hospital must also be on the same side as the main road (which bisects angle between directions to two mountains).
Solution: Draw circles of radius 10km (A) and 8km (B). Draw angle bisector of roads. The valid region is the intersection of the two circles on the correct side of the bisector.
A ship must stay at least 2km from two rocky islands. The safe zone is outside circles of radius 2km centred on each island β the complement of the union of the two circles.
Describe the locus of a point P such that angle APB = 90Β° where A and B are fixed points.
By Thales' theorem, the locus is a circle with AB as diameter (excluding points A and B).
A(0,0), B(6,0). Find the equation of the locus of points equidistant from A and B.
PA = PB β β(xΒ²+yΒ²) = β((xβ6)Β²+yΒ²)
xΒ² = xΒ²β12x+36 β 12x = 36 β x = 3 (vertical line)
1. The path traced by a moving point satisfying a given condition is called a .
2. The locus of points at a fixed distance r from a fixed point is a .
3. The locus of points equidistant from two fixed points is the .
4. The locus of points equidistant from two intersecting lines is the .
5. Points on the perpendicular bisector of AB are from A and B.
6. (4 marks) A point P moves such that it is always 5cm from a fixed point O. (a) Describe the locus of P. (b) Write the equation of this locus if O is at the origin.
7. (3 marks) A(2,3) and B(8,3). Find the equation of the locus of points equidistant from A and B.
8. (3 marks) A goat is tied to a corner of a rectangular field (30m Γ 20m) with a 15m rope. Describe and calculate the area it can graze.