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

Analysis

The question asks for the "locus of points" that maintain a constant distance from a given straight line $PQ$. In geometry, a locus is a set of all points that satisfy a specific condition or rule.

Concept

When we talk about points being "equidistant" from a line, we are looking for the path traced by a point that stays at a fixed perpendicular distance ($d$) from that line.

1. If you move along one side of the line $PQ$ at a fixed distance $d$, you trace a straight line parallel to $PQ$.
2. Because the condition does not specify "on one side only," we must consider both sides of the line.
3. Therefore, there is a parallel line at distance $d$ above $PQ$ and another parallel line at distance $d$ below $PQ$.

Solution

• Step 1: Visualize the line $PQ$ on a Cartesian plane.
• Step 2: Identify all points at a distance $d$ from $PQ$. These points form a line parallel to $PQ$ on the top side and another line parallel to $PQ$ on the bottom side.
• Step 3: Since the locus includes all such points, it consists of two lines, one on each side of $PQ$, both running parallel to the original line.
• Step 4: This collection of points is described as a "pair of parallel lines."

Conclusion

The locus of points equidistant from a fixed line is a pair of lines parallel to the original line.

Exam Tip: Always remember that a locus equidistant from a point is a circle, but a locus equidistant from a line is a pair of parallel lines. Do not confuse the two!