The perpendicular bisector of a line XY is the locus of a point

Analysis of the Question

The question asks for the definition of the perpendicular bisector of a line segment $XY$ in terms of the locus of a point. A "locus" is simply the path or set of all points that satisfy a specific geometric condition.

Explaining the Concept

In geometry, the perpendicular bisector of a line segment $XY$ is a line that:

1. Passes through the midpoint of $XY$.
2. Is perpendicular (at a $90^\circ$ angle) to $XY$.

The defining property of every point on this perpendicular bisector is that it is equidistant from the two endpoints $X$ and $Y$. If you pick any point $P$ on this bisector, the distance from $P$ to $X$ will always be equal to the distance from $P$ to $Y$ ($PX = PY$).

Step-by-Step Solution

• Option A & B: These describe a different locus (an Apollonian circle), where the ratio of distances is constant but not equal. This is not the definition of a perpendicular bisector.
• Option C: A point moving on the line $XY$ would simply be tracing the line segment itself, not its perpendicular bisector.
• Option D: This matches the geometric definition of a perpendicular bisector. By definition, the set of all points that are the same distance from two fixed points $X$ and $Y$ forms a straight line that perpendicularly bisects the segment connecting those two points.

Conclusion

The perpendicular bisector is the locus of points that are equidistant from the two fixed points $X$ and $Y$.

Exam Tip

Whenever you see the word "locus" in a JAMB question, think of it as a "rule" or "condition."

• If the rule involves being equidistant from two points, the answer is always the perpendicular bisector.
• If the rule involves being equidistant from two intersecting lines, the answer is the angle bisector.