The locus of a point which moves so that it is equidistant from two intersecting straight lines is the

Analysis

The question asks for the geometric path (locus) traced by a point that maintains an equal distance from two lines that intersect at a point.

Concept: Locus of Points

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

• When two straight lines intersect, they form angles at the point of intersection.
• If a point is equidistant from two intersecting lines, it must lie on a line that divides the angle between those two lines into two equal parts. This line is known as the angle bisector.
• Because two intersecting lines create two pairs of vertically opposite angles, there are actually two such bisectors (the internal and external angle bisectors), both of which are straight lines passing through the point of intersection.

Step-by-Step Solution

1. Visualize the intersection: Imagine two lines crossing like an 'X'.
2. Define the condition: We are looking for a point $P(x, y)$ such that the perpendicular distance from $P$ to Line 1 is equal to the perpendicular distance from $P$ to Line 2.
3. Apply the geometric property: The set of points that are equidistant from two intersecting lines are the lines that bisect the angles formed by the intersection.
4. Evaluate the options: • A "bisector of the two lines" is vague. • A "line parallel to the two lines" would not be equidistant to both unless the lines were parallel themselves (which they are not). • An "angle bisector" specifically describes the line that maintains equal distance from the two intersecting sides of an angle. • A "perpendicular bisector" refers to a line that cuts a line segment into two equal parts at a 90-degree angle, which is not the case here.

Conclusion

The locus of a point equidistant from two intersecting lines is the pair of lines that bisect the angles formed by the intersection. Among the given options, "angle bisector" is the mathematically correct term.

Exam Tip

Whenever you see the phrase "equidistant from two lines," immediately think of angle bisectors. If the question mentions "equidistant from two points," think of the perpendicular bisector. Don't confuse the two!