Test: Locus - 2 - JAMB MCQ

When two parallel lines are given, any point equidistant from both lines will lie on a line that is parallel to the given lines. This can be understood by considering the definition of parallel lines, which states that they never intersect and always maintain the same distance between them. Therefore, the locus of points equidistant from two parallel lines is a straight line parallel to those lines.

The locus of points equidistant from two given points A and B is the perpendicular bisector of the line segment AB. The perpendicular bisector is a line that intersects AB at its midpoint and forms right angles with it. Any point on the perpendicular bisector is equidistant from both A and B because it lies at an equal distance from the endpoints of the segment.

If a point moves such that its distance from a fixed point (center) is always constant, its locus is a circle. The locus of points equidistant from a fixed point forms a circle with the fixed point as its center and the constant distance as its radius. All points on the circle are at an equal distance from the fixed point.

The locus of points that are equidistant from a fixed line l and a fixed point P not on line l is a hyperbola. A hyperbola is defined as the set of points in a plane such that the difference of the distances from any point on the hyperbola to the fixed point (focus) and the fixed line (directrix) is constant. In this case, the fixed point P acts as the focus and the fixed line l acts as the directrix.

The locus of points whose distances from two fixed points A and B are in a constant ratio is a hyperbola. In this case, the hyperbola has foci at points A and B. The ratio of 2:3 indicates that the difference of distances from any point on the hyperbola to focus A and focus B will always be in the ratio of 2:3.

If a point moves in a plane such that the sum of its distances from two fixed points A and B is constant, the locus of this point is an ellipse. An ellipse is defined as the set of points in a plane where the sum of the distances from any point on the ellipse to two fixed points (foci) is constant. In this case, the sum of distances from the moving point to fixed points A and B remains constant, resulting in an elliptical locus.

The locus of points equidistant from the endpoints of a line segment AB is the perpendicular bisector of AB. The perpendicular bisector is a line that intersects AB at its midpoint and forms right angles with it. Any point on the perpendicular bisector is equidistant from both endpoints of the line segment because it lies at an equal distance from each.

The locus of points that are equidistant from the x-axis and the y-axis is a circle with the center at the origin. The points on the circle will have equal distances from both the x-axis and the y-axis. Since the origin is equidistant from the x-axis and the y-axis, it serves as the center of the circle.

The locus of the midpoint of a chord of a circle with center O and radius r, which subtends a constant angle at the center, is an ellipse. As the chord rotates around the circle, the midpoint of the chord traces out an ellipse. The major axis of the ellipse coincides with the diameter of the circle that subtends the constant angle at the center.

The locus of points that are equidistant from two intersecting lines l1 and l2 is the angle bisector of the angle between them. The angle bisector is a line that divides the angle formed by the intersecting lines into two equal angles. Any point on the angle bisector is equidistant from both lines, as the distances to the lines are equal due to the symmetry of the angle bisector.