Loci
How to describe a locus in words and find its equation, including perpendicular bisectors and circles. It is a short unit that appears in coordinate geometry questions.
What you will learn
- 12.1 What a locus is.
- 12.2 Describe and sketch the locus of points that:
- 12.3 Find equations of simple loci: straight lines, circles and parabolas (y = ax² + bx + c).
You need first: Coordinates, Equations of straight lines, Quadrilaterals and centres of triangles
Key terms
| English | 中文 |
|---|---|
| locus (plural loci) | 軌跡 |
| moving point | 動點 |
| equidistant | 等距 |
| perpendicular bisector | 垂直平分線 |
| angle bisector | 角平分線 |
| parabola | 拋物線 |
Standard steps
The five pictures
| Condition | Locus |
|---|---|
| distance r from a point C | circle, centre C, radius r |
| equidistant from A and B | perpendicular bisector of AB |
| distance d from a line | two lines parallel to it, one on each side |
| equidistant from two parallel lines | the parallel line halfway between |
| equidistant from two intersecting lines | the two angle bisectors |
Find the equation of a locus
- Let the moving point be P(x, y).
- Write the condition in words, then with the distance formula.
- Square both sides to remove the roots.
- Expand and simplify.
- Name it ("the locus is a straight line / circle / parabola") and describe it if asked.
"Describe the geometric relationship"
Say what the locus is and how it relates to the given objects: "the perpendicular bisector of AB", not just "a straight line".
- Name the shape: a straight line, a circle or a parabola.
- Name how it relates to the given points or lines: "the perpendicular bisector of AB", "the circle with centre C and radius 3".
- Check with one point: the mid-point of AB, or the centre, should fit the equation or the description.
Worked examples
PA = PB
(x − 1)² + (y − 2)² = (x − 5)² + (y − 6)²
−2x + 1 − 4y + 4 = −10x + 25 − 12y + 36
8x + 8y − 56 = 0
x + y − 7 = 0Check: the mid-point of AB, (3, 4), satisfies it, and the slope −1 is perpendicular to AB's slope 1. It is the perpendicular bisector of AB.
√[x² + (y − 1)²] = |y + 1|
x² + y² − 2y + 1 = y² + 2y + 1
x² = 4y, i.e. y = x²/4: a parabolaCommon mistakes
- Forgetting the second line when the locus is "distance d from a line".
- Giving only one angle bisector for two intersecting lines.
- Not squaring both sides fully: (x − 5)² expanded as x² − 25.
- Describing the locus as "a line" without saying which line.
How the DSE tests it
- A(2): find the equation of a locus, describe it geometrically, then use it (for example, find where it meets an axis or the area of a figure).
- Section B: a circle locus, then an intersection with a line.
- Paper 2: identify the locus from a description or sketch.
Questions you might not ask in class
What exactly is a locus?
The path a point traces when it must obey a rule.
Why do we square both sides?
The distance formula has square roots. Squaring removes them, and because distances are never negative, no answers are lost.
Do I need to name the shape?
Yes, if the question says "describe". Name it and say how it relates to the given points or lines.