Chapter Four: Locus – Additional Mathematics Form Two
The movement of an object from one point to another can be described by its location. The path traced by a moving object which satisfies stated set of conditions is called locus. For instance, the earth revolves around the sun, the centre of the wheel of a vehicle moving on a straight road, and the lines drawn on the middle of the road each forms locus.
In this chapter, you will learn about a locus about a fixed point, two fixed points, a line, and two intersecting lines. The competencies developed will help you in construction of bridges, flyovers, roads, sports and games run ways in the field of engineering, among many other applications.
Locus About a Fixed Point
A locus is defined as a path traced by a moving object. Geometrically, it is a set of all points whose location is determined by one or more specified conditions. For instance, a line, a circle, an ellipse, a parabola, and many other shapes are described by the loci of points. The plural of locus is loci and it is derived from the word “location”.
Geometrical Representation
When an object moves such that, it is always at a constant distance from a fixed point, its locus is described as a circle.
Fixed Point (O): Center of the circle
Moving Point (P): Any point on the circumference
Constant Distance: Radius (OP)
Activity 1: Constructing a locus about a fixed point
Individually or in a group, perform the following tasks:
- Locate a fixed point O on a graph paper, and a point P at a distance of your choice from a fixed point O to represent a moving object.
- Put the end point of the compass at a point O and its pencil end at P.
- Construct an arc by moving the pencil end away from point P.
- Continue drawing an arc until the pencil end of the compass meets the point P again.
- Describe the name of the figure obtained.
- Repeat using different distances of point P from a fixed point O.
- Share your results with other students through discussion for more inputs.
Example
Construct the locus of a point Q at a constant distance of 3 cm from a fixed point E.
Solution
The locus of a point Q is a circle with centre at a fixed point E and radius 3 cm.
Construction:
1. Mark point E as the center
2. Set compass to 3 cm radius
3. Draw a complete circle around E
4. All points on this circle are 3 cm from E
Cartesian Representation of Locus About a Fixed Point
Equation of a circle: (x – a)² + (y – b)² = r²
Where (a, b) is the center and r is the radius
Example
Find the equation of the locus of the point P(x, y) which moves on a plane such that its distance from the origin is always 3 units.
Solution
The locus of a moving point P(x, y) is a circle with centre at the origin and radius r units.
Equation: x² + y² = r²
Substitute r = 3: x² + y² = 9
Therefore, the equation of the locus is x² + y² = 9
Example
A point is moving on the xy-plane such that its distance from a fixed point (3, 4) is always 5 units. Find the equation of the locus.
Solution
Using the formula: (x – a)² + (y – b)² = r²
Where (a, b) = (3, 4) and r = 5
(x – 3)² + (y – 4)² = 25
Expanding: x² – 6x + 9 + y² – 8y + 16 = 25
Therefore, x² + y² – 6x – 8y = 0
Locus About Two Fixed Points
Perpendicular Bisector
The locus of a point which moves such that it is equidistant from two fixed points A and B is the perpendicular bisector of the line joining the two points.
Construction Steps
- Locate two points A and B at a reasonable distance apart
- Draw line segment AB connecting the points
- Adjust compass to obtain a length more than half of AB
- Place compass pointer at A and draw arcs above and below AB
- Maintaining the radius, place compass pointer at B and draw arcs to cross the previous arcs
- Mark the intersection points as X and Y
- Join points X and Y with a straight line
- This line XY is the perpendicular bisector
Example
Construct the locus of a point P which is equidistant from the two fixed points A and B such that AB = 8 cm.
Solution
The locus is the perpendicular bisector of the line AB.
Construction:
1. Draw line AB = 8 cm
2. Find midpoint of AB
3. Construct perpendicular line through midpoint
4. This perpendicular line is the required locus
Cartesian Representation of Locus About Two Fixed Points
Distance Formula: d = √[(x₂ – x₁)² + (y₂ – y₁)²]
Example
A point E moves so that it is equidistant from the points M(0, 5) and N(2, 7). Show that the equation of the locus of E is x + y – 7 = 0.
Solution
Let E be the point (x, y). Since point E is equidistant from M and N:
EM = EN
√[(x – 0)² + (y – 5)²] = √[(x – 2)² + (y – 7)²]
Squaring both sides:
(x – 0)² + (y – 5)² = (x – 2)² + (y – 7)²
x² + y² – 10y + 25 = x² – 4x + 4 + y² – 14y + 49
Simplifying: 4x + 4y – 28 = 0
Therefore: x + y – 7 = 0
Locus About a Line
Parallel Lines as Locus
The locus of a point moving such that it is always equidistant from a straight line is a pair of lines parallel to the straight line.
Example
Construct the locus of a point P that moves at a constant distance of 2 cm from a straight line AB.
Solution
The locus is a pair of parallel lines 2 cm from the line AB.
Construction:
1. Draw line AB
2. Using a set square, construct lines parallel to AB at 2 cm distance on both sides
3. These parallel lines form the required locus
Locus About Two Intersecting Lines
Angle Bisectors
The locus of a point which is equidistant from two intersecting straight lines consists of a pair of straight lines which bisect the angles between the two given lines.
Example
Construct the locus of a point P such that it is always equidistant from two intersecting lines AB and CD.
Solution
The locus is the angle bisectors of the angles formed between intersecting lines AB and CD.
Construction:
1. Draw two intersecting lines AB and CD
2. Construct the angle bisectors of all four angles formed
3. These bisectors form the required locus
Exercise 1
- The student radio station has a broadcasting range of 4 km. Describe the locus of points which represents the outer edge of the broadcasting range.
- Two points R and P are moving such that they are always 2.5 cm and 4 cm, respectively from the same fixed point O. Construct the loci of the points R and P.
- Find the equation of the locus of a point A(x, y) which moves so that it is 4 units from the origin.
- Find the equation of the locus of a point P(x, y) which is moving so that it is always 3 units from the point (-2, -3).
Chapter Summary
- A locus is defined as a path traced by a moving point.
- The locus of a point moving such that it is always at constant distance from a fixed point is a circle.
- A point moving such that it is always equidistant from two fixed points forms a perpendicular bisector of the line joining the two fixed points.
- The distance between two fixed points P(x₁, y₁) and Q(x₂, y₂) is given by d = √[(x₂ – x₁)² + (y₂ – y₁)²].
- The locus of a point moving such that it is always at a certain distance from a straight line is a pair of parallel lines.
- The locus of a point moving such that it is always equidistant from two intersecting lines is the bisector of the angle between the two lines.
Revision Exercise
- Construct the locus of a moving point such that it is always 5cm from a fixed point.
- Find the equation of the locus of a point C which moves such that it is k units from the origin.
- If a point A is at unit distance from the point (-1,2). Find the locus of the point A in Cartesian form.
- Two buildings are 2 km apart. The canal is to be dug such that the distance from any point on the canal to each building is always the same. Describe where the canal should be dug.
- Find the equation of the locus of a point which is equidistant from points (1,2) and (3,4).