Combining two loci
Many problems ask for points that satisfy two conditions at once. Each condition gives its own locus; the points you want lie where the two loci intersect. Draw both loci accurately, then mark every crossing point.
Key idea
Two circles can cross at 0, 1 or 2 points. A line can cut a circle at 0, 1 or 2 points. The final answer is the intersection of the two loci.
Counting intersection points
Consider points that are 5 cm from A and also 5 cm from B, where A and B are 8 cm apart. Each condition is a circle of radius 5 cm. Since the radii (5 + 5 = 10 cm) exceed the gap AB = 8 cm, the two circles overlap and meet at 2 points. Those two points are the required locus.
Worked example
A and B are 8 cm apart. Find how many points are exactly 5 cm from A and 5 cm from B.
Locus 1: circle centre A, radius 5 cm. Locus 2: circle centre B, radius 5 cm. Sum of radii = 5 + 5 = 10 cm > 8 cm, and the difference 0 < 8, so the circles intersect at 2 points.
Regions
Sometimes a condition is 'less than' a distance, giving a region (inside a circle, or the strip between two lines). The answer is then the overlap of the two regions.
Remember
- Two conditions → intersection of two loci.
- Circle–circle meets at 0, 1 or 2 points; compare the sum of radii with the centre distance.
- 'At most' or 'within' means shade a region, not just a line.