The Locus of a Moving Point

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.

INTERSECTION OF TWO LOCI

Sometimes you may be required to determine the locus of a point that satisfies two 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.

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