Definition
A circle is the locus of points at a fixed distance from a fixed point
Geometry • Circle • Locus
Statement
The locus of a point that moves so that its distance from a fixed point is a constant is the circle with centre and radius . Naming a locus this way needs both directions: every point of the locus is at distance from , and every point at distance from belongs to the locus.
How to use it
- 1Show that the moving point keeps a constant distance from one fixed point and the locus is identified at once, with no coordinates needed.
- 2Check the converse too, or you may call the locus a circle when it is only an arc of one.
- 3Two such loci about the same centre with different radii are concentric circles, and the region between them is an annulus.
Worked example
A point moves so that its distance from a fixed point is always cm. Describe the path of and find its length. (Use .)
Step-by-step
- 1cm for every position of , so traces the circle of centre and radius cm
Why: A constant distance from one fixed point is exactly the definition of a circle.
- 2length cm
Answer
The circle of radius cm about , of length cm
Watch out
The locus is the whole circle only when the moving point is free to go right round; a restriction leaves an arc. A constant distance from a fixed LINE is a different locus — a pair of parallel lines, not a circle.