locus is a set of points which satisfy certain geometric conditions. Many geometric shapes are most naturally and easily described as locus. For example, a circle is the set of points in a plane which are a fixed distance from a given point the center of the circle.

Example: A locus


The following diagrams give the locus of a point that satisfy some conditions.

Circle with Center

Locus of points equidistant from a point A will form a circle with center A.

Two Lines

Locus of points that are equidistant from two lines will be bisect the angle formed by the two lines.

Line Segment

Locus of points equidistant from a line segment.

Perpendicular Bisector

Locus of points equidistant from two points A and B forms a perpendicular bisector of the line AB.

A step-by-step procedure for finding plane locus:

Step 1: If possible, choose a coordinate system that will make computations and equations as simple as possible.

Step 2: Write the given conditions in mathematical form involving the coordinates.

Step 3: Simplify the resulting equation.

Step 4: Identify the shape cut out by the equation.

1. Find the point on the x-axis, which is equidistant from (7, 6) and (-3, 4).

Sol: Let A (7, 6), B (-3, 4)

Let P(x, 0) be the point on x-axis which is equidistant from A and B.

Then PA = PB ⇒ PA² = PB²

⇒ (x – 7)² + (0 – 6)² = (x + 3)² + (0 – 4)²

⇒ x² – 14x + 49 + 36 = x² + 6x + 9 + 16

⇒ -14x + 85 = 6x + 25 ⇒ 20x = 60

⇒ x = 3

The required point is P (3, 0).