What is the loci of a distance 4cm from a given point P?

Analysis

The question asks for the "locus" (the path traced by a point) that maintains a constant distance of 4cm from a fixed point $P$.

Concept Explanation

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

• When you have a fixed point (let's call it the center) and you look for all points that are a fixed distance (the radius) away from it, you are describing a circle.
• Imagine holding one end of a 4cm string at point $P$ and moving the other end around in every possible direction while keeping the string taut. The path traced by the tip of the string will be a circle with a radius of 4cm.

Step-by-Step Solution

1. Identify the fixed point: Point $P$ acts as the center of the shape.
2. Identify the fixed distance: The distance of 4cm acts as the radius ($r = 4\text{cm}$).
3. Apply the definition: By definition, a circle is the locus of points in a plane that are at a constant distance (radius) from a fixed point (center).
4. Evaluate the options: • A straight line would not maintain a constant distance from a single point in all directions. • A circle of radius 4cm perfectly matches our definition. • "Perpendicular" describes an angle, not a locus of points. • A circle of diameter 4cm would have a radius of only 2cm, which is incorrect.

Conclusion

The locus of points at a constant distance from a fixed point is a circle where the fixed point is the center and the constant distance is the radius. Therefore, the locus is a circle of radius 4cm.

Exam Tip: Whenever you see the word "locus" in a JAMB question, think of it as a "rule" or "path." If the rule involves a fixed distance from a point, it is always a circle. If it involves a fixed distance from a line, it is a pair of parallel lines.