Exercise solution 3, 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. For example, if a question asks for all points a fixed distance from one point, you must draw every point that matches that distance, not just one or two examples.

B) All points that are the same distance from a single point form a circle. The fixed point is the centre of the circle, and the fixed distance is the radius. So if all points are 5 cm from a point \(P\), the locus is a circle with centre \(P\) and radius \(5\text{ cm}\).

C) It is important to distinguish between radius and diameter. The radius goes from the centre to the circle, while the diameter goes all the way across the circle through the centre. They are linked by the formula \(d=2r\). For example, if the radius is \(3.5\text{ cm}\), then the diameter is \(7\text{ cm}\).

D) To construct this type of locus accurately, use a compass. Place the compass point on the given centre, open it to the required radius using a ruler, and draw the full circle without changing the compass width. Accuracy matters: if the compass opening changes, the locus will be incorrect.

E) A point is on the locus only if it is exactly the required distance from the centre. Points inside the circle are less than that distance, and points outside the circle are greater than that distance. For instance, on a circle of radius \(6\text{ cm}\), any point \(4\text{ cm}\) from the centre is inside the circle, while any point \(8\text{ cm}\) from the centre is outside it.

F) In locus questions, the wording tells you what shape to expect. “Points a fixed distance from one point” means a circle. This is a key fact to recognise quickly, because it tells you both the shape to draw and how to draw it.