The Locus of a Moving Point



In this lesson, we will learn

  • the rules of the Locus Theorem
  • how the rules of the Locus Theorem can be used in real world examples.
  • how to determine the locus of points that will satisfy more than one condition.

When a point moves in a plane according to some given conditions the path along which it moves is called a locus. (Plural of locus is loci.).

CONDITION 1 :

A point P moves such that it is always m units from the point Q

Locus formed: A circle with centre Q and radius m.

 

 


Example :

Construct the locus of a point P at a constant distance of 2 cm from a fixed point Q.

Solution:

Construct a circle with centre Q and radius 2 cm.

 



CONDITION 2
:

A point P moves such that it is equidistant form two fixed points X and Y

Locus formed: A perpendicular bisector of the line XY.

 


Example:

Construct the locus of point P moving equidistant from fixed points X and Y and XY = 6 cm.

Solution:

Construct a perpendicular bisector of the line XY.

 


CONDITION 3:


A point P moves so that it is always m units from a straight line AB

Locus formed: A pair of parallel lines m units from AB.

 

 


Example:

Construct the locus of a point P that moves a constant distant of 2 cm from a straight line AB.

Solution:

Construct a pair of parallel lines 2 cm from AB.

 



CONDITION 4:

A point P moves so that it is always equidistant from two intersecting lines AB and CD

Locus formed: Angle bisectors of angles between lines AB and CD.

Example:

The following figure shows two straight lines AB and CD intersecting at point O. Construct the locus of point P such that it is always equidistant from AB and CD.

 


Solution:

Construct angles bisectors of angles between lines AB and CD.

 





 The following video explains the five rules of locus theorem.

The following video will explain the five rules of locus theorem using real world examples.




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Intersection of two loci

Sometimes you may be required to determine the locus of a point that satisfies two or more conditions. We could do this by constructing the locus for each of the conditions and then determine where the two loci intersect.

Example :

Given the line AB and the point Q, find one or more points that are 3 cm from AB and 5 cm from Q.

 

 

Solution:

Construct a pair of parallel lines 3 cm from line AB. Draw a circle with centre Q and radius 5 cm.

 


The points of intersections are indicated by points X and Y.

It means that the locus consists of the two points X and Y.

Example:

Given a square PQRS with sides 3 cm. Construct the locus of a point which is 2 cm from P and equidistant from PQ and PS. Mark the points as A and B.

Solution:

Construct a circle with centre P and radius 2 cm. Since PQRS is a square the diagonal PR would be the angle bisector of the angle formed by the lines PQ and PS. The diagonal when extended intersects the circle at points A and B

 

Note: A common mistake is to identify only one point when there could be another point which could be found by extending the construction lines or arcs; as in the above examples.





Intersecting Loci




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