Loci
Introduction
A locus (plural: loci) is the set of all points that satisfy a given condition. In geometry, understanding loci helps you determine the path traced by a point moving according to a specific rule. In ICSE Class 10, you learn fundamental loci and use them in constructions and proofs.
Definition
A locus is the path traced by a moving point under a given condition, or the set of all points that satisfy a specific geometric condition.
Important Loci
Locus 1: Points at a Fixed Distance from a Fixed Point
The locus of points at a constant distance d from a fixed point O is a circle with centre O and radius d.
- If the point moves in a plane: the locus is a circle.
- If the point moves in space: the locus is a sphere.
Locus 2: Points Equidistant from Two Fixed Points
The locus of points equidistant from two fixed points A and B is the perpendicular bisector of AB.
Every point on the perpendicular bisector of AB is equidistant from A and B.
Locus 3: Points Equidistant from Two Intersecting Lines
The locus of points equidistant from two intersecting lines is the pair of angle bisectors of the angles formed by the lines.
If two lines intersect at an angle, there are two angle bisectors (internal and external), and every point on either bisector is equidistant from both lines.
Locus 4: Points at a Fixed Distance from a Line
The locus of points at a fixed distance d from a given line L is a pair of lines parallel to L at distance d on either side.
Locus 5: Points Equidistant from Two Parallel Lines
The locus of points equidistant from two parallel lines is a line parallel to both lines and midway between them.
Intersection of Loci
A point may need to satisfy two conditions simultaneously. The point(s) of intersection of the two loci give the required point(s).
Example: Find a point P that is 3 cm from A and 2 cm from B.
- Step 1: Draw a circle with centre A, radius 3 cm.
- Step 2: Draw a circle with centre B, radius 2 cm.
- Step 3: The intersection points of the two circles are the required points.
Worked Examples
Example 1: Locus of Points
Draw and describe the locus of a point that moves such that its distance from a fixed line AB is always 2 cm.
Solution: The locus is a pair of lines parallel to AB at a distance of 2 cm on either side. (If the condition specifies one side, it is a single parallel line.)
Example 2: Intersection of Loci
Construct a point P inside triangle ABC such that P is equidistant from AB and AC and also 3 cm from BC.
Solution: P must satisfy two conditions:
- Equidistant from AB and AC → P lies on the angle bisector of ∠A.
- 3 cm from BC → P lies on a line parallel to BC at a distance of 3 cm.
The intersection of the angle bisector with the parallel line gives the required point P.
Example 3: Four Centres of a Triangle
Locus concepts are used to define the four centres of a triangle:
| Centre | Locus Intersection | Description |
|---|---|---|
| Circumcentre | Perpendicular bisectors of sides | Equidistant from vertices |
| Incentre | Angle bisectors | Equidistant from sides |
| Orthocentre | Altitudes | No equal-distance property |
| Centroid | Medians | Divides medians in 2 : 1 ratio |
Note: The circumcentre is the locus of points equidistant from the three vertices. The incentre is the locus of points equidistant from the three sides.
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| Calling a circle the locus for a fixed point condition | Correct: circle with centre at the fixed point |
| Confusing perpendicular bisector with angle bisector | Perp bisector: equidistant from TWO POINTS. Angle bisector: equidistant from TWO LINES |
| Drawing only one line for locus from a line | Two parallel lines (unless specified "on one side") |
| Thinking all points on a circle satisfy any fixed-distance condition | Only if the fixed point is the centre |
ICSE Exam Focus
Loci carry 4–6 marks in ICSE exams. Questions include:
- Describing the locus for a given condition.
- Constructing a point satisfying two locus conditions.
- Using locus properties in geometric constructions.
- Simple proof-based locus problems.
Marks Blueprint:
| Topic | Marks |
|---|---|
| Describing a locus | 2 |
| Construction with two loci | 3 |
| Locus in triangle geometry | 2 |
| Proof-based locus question | 2 |
Self-Test Questions
-
Describe the locus of a point that moves such that it is always at a constant distance of 5 cm from a fixed point O.
-
Construct a point equidistant from two given points A and B and also 4 cm from a third point C.
-
Describe the locus of the vertex A of triangle ABC if base BC is fixed and area remains constant.
-
Explain why the circumcentre is the point of concurrency of perpendicular bisectors. What is the locus property being used?
-
A point moves so that it is always equidistant from two intersecting lines. Describe its locus.
In ICSE, locus questions commonly combine with constructions. Use a sharp pencil and compass for accuracy — marks are awarded for neatness and precision.
