Exercise solution 2, page 89 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) A locus is the set of all points that satisfy a given condition. In these questions, you are not finding one point only: you are describing every point that works. The answer may be a point, a line, a curve, or an area.

B) When points are a fixed distance from one point, the locus is a circle. A circle is the set of all points at the same distance from its centre. For example, all points \(5\) km from a lighthouse form a circle of radius \(5\) km centred on the lighthouse.

C) If the condition says within a distance, this describes an area, not just a boundary. For example, points within \(4\) m of a tree include all points whose distance from the tree is less than or equal to \(4\) m, so the locus is the disc inside the circle of radius \(4\) m.

D) When points are a fixed distance from a straight line segment, the shape is found by imagining all points that stay that distance from the segment. This usually gives a strip beside the segment, together with curved ends. If the distance is between two values, such as more than one distance but less than another, the locus is the region between two such boundaries.

E) The phrase equidistant from two lines means the point is the same perpendicular distance from each line. If the two lines are parallel, the locus is the line exactly halfway between them. For example, between two parallel roads, the points equidistant from both roads lie on the middle line running parallel to them.

F) The phrase equidistant from two points means the point is the same distance from each point. The locus is the perpendicular bisector of the segment joining the two points. This is the line that cuts the segment into two equal parts at \(90^\circ\).

G) To be equidistant from four corners of a rectangle, a point must be at the same distance from opposite corners and also from the other pair of opposite corners. So you can think of this as the point where the perpendicular bisectors meet. In a rectangle, that is the centre of the rectangle.

H) Be careful with the wording:

“from a point” usually suggests a circle if the distance is fixed.

“from a line” often gives a strip or band around the line.

“equidistant from two parallel lines” gives a middle line.

“equidistant from several corners” may give a single point if all conditions must be true at once.

I) It is also important to notice whether the question describes a boundary or a region:

If the distance is exactly a value, such as “\(7\) km from”, the locus is a boundary.

If the distance is less than, greater than, or between two values, the locus is usually an area.

For instance, points more than \(3\) m but less than \(6\) m from a post form the ring-shaped region between two circles.