The locus of points equidistant from a fixed point.

1. Analyze the question

The question asks for the geometric shape formed by all points that maintain a constant distance (equidistant) from a single, fixed point in a two-dimensional plane.

2. Explain the concept

In coordinate geometry, a locus is a set of all points that satisfy a specific condition.

• A fixed point is referred to as the center.
• The constant distance from that center is referred to as the radius.

By definition, a circle is the set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the center). If you imagine a compass used in your technical drawing set, the needle stays at the fixed point while the pencil traces the locus of points equidistant from that needle.

3. Show the solution

• Let the fixed point be $O(h, k)$.
• Let any point on the locus be $P(x, y)$.
• The condition "equidistant" means the distance $d$ between $P$ and $O$ is constant ($r$).
• Using the distance formula: $\sqrt{(x - h)^2 + (y - k)^2} = r$
• Squaring both sides gives: $(x - h)^2 + (y - k)^2 = r^2$
• This is the standard equation of a circle.

4. Identify the answer

Comparing our definition and the mathematical derivation to the options provided:

• A. Circle: This matches our definition perfectly.
• B. Perpendicular lines: These do not maintain a constant distance from a single point.
• C. Straight line: This represents points equidistant from two fixed points (the perpendicular bisector), not one.
• D. Bisector: This is a line that divides an angle or segment into two equal parts.

The correct option is A.

5. Exam Tip

Whenever you see the word "locus" in JAMB Mathematics, think of it as a "path" or "shape."

• Equidistant from one point = Circle.
• Equidistant from two points = Perpendicular bisector of the line joining them.
• Equidistant from two intersecting lines = Angle bisector.