Loci in Two Dimensions

In geometry, a locus (plural: loci) is a set of points whose location is satisfied by or determined by one or more specified conditions. In simpler terms, it is the path traced by a point moving according to a specific rule.

1. Basic Loci and Geometric Interpretations

• Locus of points at a constant distance from a fixed point: This is a circle. The fixed point is the center, and the constant distance is the radius.
• Locus of points equidistant from two fixed points: This is the perpendicular bisector of the line segment joining the two points.
• Locus of points equidistant from two intersecting straight lines: This consists of the angle bisectors of the angles formed by the lines. There are usually two such lines, perpendicular to each other.
• Locus of points at a constant distance from a fixed straight line: This consists of two parallel lines, one on each side of the given line.
• Locus of points equidistant from two parallel lines: This is a third parallel line located exactly midway between the two given lines.

2. Coordinate Geometry of Loci

In coordinate geometry, we represent loci using algebraic equations. For example:

• Circle: The locus of a point $(x, y)$ at a distance $r$ from $(h, k)$ is $(x - h)^2 + (y - k)^2 = r^2$.
• Perpendicular Bisector: If points are $A(x_1, y_1)$ and $B(x_2, y_2)$, any point $P(x, y)$ on the locus satisfies $PA = PB$. We use the distance formula: $\sqrt{(x-x_1)^2 + (y-y_1)^2} = \sqrt{(x-x_2)^2 + (y-y_2)^2}$.

3. Practical Applications in JAMB

JAMB often tests the ability to identify the shape or the specific equation of a locus. You must visualize the movement: if a goat is tied to a pole with a rope of 5m, the locus of the goat's movement is a circle with radius 5m. If the goat is tied to a long straight wire and can move 2m away from it, the locus is a pair of parallel lines.