The locus of a point equidistant from two intersecting lines is

Analysis

The question asks for the geometric definition of a locus (a set of points satisfying a specific condition) that remains at an equal distance from two lines that intersect at a point.

Concept Explanation

In geometry, when two lines intersect, they form four angles. If you want to find a point that is equidistant from both lines, you are looking for a path that splits these angles exactly in half.

1. Angle Bisector Theorem: Any point lying on the bisector of an angle is equidistant from the two arms of that angle.
2. Intersecting Lines: Because two intersecting lines create two pairs of vertically opposite angles, there are two bisectors: • The internal bisector (which splits the acute/obtuse angles). • The external bisector (which is perpendicular to the internal bisector).
3. Together, these two bisectors form a pair of perpendicular lines that pass through the point of intersection. Any point on either of these lines will maintain the same perpendicular distance to both original intersecting lines.

Step-by-Step Solution

1. Let the two intersecting lines be $L_1$ and $L_2$.
2. A point $P(x, y)$ is equidistant from $L_1$ and $L_2$ if the perpendicular distance $d_1$ from $P$ to $L_1$ equals the perpendicular distance $d_2$ from $P$ to $L_2$ (i.e., $|d_1| = |d_2|$).
3. The equation $|d_1| = |d_2|$ mathematically expands into $d_1 = d_2$ or $d_1 = -d_2$.
4. These two equations represent the two straight lines that bisect the angles formed by $L_1$ and $L_2$.
5. Therefore, the locus is the pair of angle bisectors.

Evaluating the Options

• Option A describes an ellipse.
• Option B describes a parabola.
• Option C is incorrect because a perpendicular bisector applies to a line segment, not two intersecting lines.
• Option D correctly identifies the pair of bisectors of the angles between the two lines.

Exam Tip

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

• Equidistant from a point and a line = Parabola.
• Equidistant from two points = Perpendicular bisector.
• Equidistant from two intersecting lines = Pair of angle bisectors.