The locus of a point which is equidistant from the line PQ forms a

Analysis

The question asks for the locus (the set of all points that satisfy a specific condition) of a point that maintains a constant distance from a fixed line $PQ$.

Concept Explanation

In geometry, the locus of points equidistant from a fixed line is a pair of lines running parallel to the original line.

Think of it this way: if you have a straight road (line $PQ$), and you want to walk along a path where you are always exactly 5 meters away from that road, you would have to walk in a straight line parallel to the road. Because you could be 5 meters to the "left" or 5 meters to the "right" of the road, there are two such parallel lines—one on each side.

Step-by-Step Solution

1. Define the fixed object: We have a line segment $PQ$.
2. Define the condition: The distance from any point $(x, y)$ on the locus to the line $PQ$ must be a constant, let's call it $d$.
3. Visualize the path: • If you move along a path that is always distance $d$ from the line $PQ$, you are tracing a line parallel to $PQ$. • Since the distance can be measured on either side of the line $PQ$, there are two such lines, one on each side, both parallel to $PQ$.
4. Evaluate the options: • A & C: A circle is the locus of points equidistant from a fixed point (the center), not a line. • D: A perpendicular line would move closer to or further from the line $PQ$ as you travel along it, so it cannot be the locus. • B: This correctly describes the two parallel lines (one on each side) that maintain a constant distance from the original line $PQ$.

Conclusion

The locus of a point equidistant from a fixed line consists of two lines parallel to the given line, one on each side.

Exam Tip: Always distinguish between a fixed point and a fixed line.

• Equidistant from a point = Circle.
• Equidistant from a line = Pair of parallel lines.
• Equidistant from two points = Perpendicular bisector.