What is a locus?
A locus is the set of all points that satisfy one given condition. Instead of a few scattered points, we describe the whole path or shape they form. Reading the condition carefully tells you the shape.
Key idea
- Fixed distance r from a point → a circle of radius r.
- Fixed distance d from a straight line → two parallel lines, one on each side.
- Equidistant from two points → the perpendicular bisector of the segment joining them.
- Equidistant from two lines that cross → the angle bisector.
Distances from a point
If a point moves so that it is always exactly 5 cm from a fixed point O, it traces a circle of radius 5 cm centred at O. Every point on that circle is 5 cm from O, and no other point is.
Worked example
Find the locus of points 7 cm from a fixed point O, and the area it encloses. (Take π = 22/7.)
The locus is a circle of radius 7 cm centred at O. Area enclosed = π r² = (22/7) × 7 × 7 = 22 × 7 = 154 cm².
Equidistant conditions
Points the same distance from two fixed points A and B lie on the perpendicular bisector of AB. For A(0, 0) and B(6, 0) the midpoint is (3, 0), so the locus is the vertical line x = 3.
Remember
- Translate the words into a shape before drawing.
- 'From a point' usually means a circle; 'from a line' means parallel lines.
- 'Equidistant' means a bisector — of a segment or of an angle.