Loci and Regions

Grade 5

What is a locus?

A locus (plural: loci) is the set of all points that satisfy a given condition. Think of it as the path traced by a point that follows a rule.

The four standard loci at GCSE: (1) a fixed distance from a point → a circle; (2) a fixed distance from a line → two parallel lines and semicircles at ends; (3) equidistant from two points → perpendicular bisector; (4) equidistant from two lines → angle bisector.

Combining loci into regions

Many questions ask you to shade a region — the intersection of two or more loci conditions. Draw each boundary first, then shade where both conditions are met.

Read "within 3 cm of point A" as: shade the inside of the circle of radius 3 cm centred on A. Read "closer to line AB than line BC" as: shade the side of the angle bisector nearer AB.

Worked example

EXAMPLE

On a scale diagram, shade the region that is less than 4 cm from point P and closer to A than to B.

Draw a circle of radius 4 cm centred on P — points inside satisfy the first condition. Construct the perpendicular bisector of AB — points on A's side satisfy the second. The required region is inside the circle and on the A-side of the perpendicular bisector (the intersection of the two regions).

Try this one

A point \( P \) moves so that it is always exactly \( 4 \) cm from a fixed point \( O \).

Describe fully the locus of the point \( P \). [2]

Nova's hint:

This is about the locus of points a fixed distance from a single point.

Ask yourself: 'if I stay the same distance from one point in every direction, what shape do I trace out?' Think about what all those points have in common.

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