Loci (Locus and Its Constructions)
320. The Moving Trail · Discovering the path of a point
The distance from O to P is always equal to r.
A locus is the path traced by a point that obeys a rule. A point that keeps a fixed distance r from a fixed point O traces a circle of radius r centred at O.
What this lesson covers
Try to break it
Drag P around. The constraint keeps P on the circle, and OP stays fixed at r at every position. That single rule — "stay at distance r from O" — IS the circle. The locus of points satisfying that rule traces out the full circle and nothing else.
How you build it
Trace the locus of a moving point.
- Place the fixed point O — the centre of the locus.
- Mark a point P at any distance r from O — this is the fixed distance.
- With centre O and radius OP, draw the arc. Every point on this arc sits exactly distance r from O — that's the LOCUS of points at distance r from O: a circle.
The proof, step by step
Prove that the locus of points at a fixed distance from O is a circle.
- A locus is the path traced by a moving point.
- The point must satisfy a specific condition.
- The locus is the set of all such points.
Worked example
The locus of a point that moves at a constant distance from a fixed point is a:
By definition, the locus of a point moving at a constant distance from a fixed point is a circle.
- A straight line
- A circle — correct
- A parabola
- An ellipse