A locus question gives a rule for a moving point P(x, y) and asks for an equation that every such point satisfies. The recipe is the same each time: write the rule with distances, square, substitute and simplify.
This lesson is part of SPM Additional Mathematics coordinate geometry. It relies on the distance formula from using gradient, midpoint and distance together.
The four-step recipe
- Write the rule in words, using PA, PB and so on for distances.
- Turn it into an equation and square both sides to remove roots.
- Replace each squared distance with (x − a)² + (y − b)².
- Expand and simplify to one equation in x and y.
Always finish with a check point. It takes thirty seconds and catches most errors.
Worked example 1: equidistant from two points
P moves so that PA = PB, where A(1, 2) and B(5, 4).
Square both sides: PA² = PB². Substitute: (x − 1)² + (y − 2)² = (x − 5)² + (y − 4)².
Expand the left side: x² − 2x + 1 + y² − 4y + 4. Expand the right side: x² − 10x + 25 + y² − 8y + 16.
Subtract: 8x + 4y − 36 = 0, so 2x + y − 9 = 0.
Check: the midpoint of AB is (3, 3), and 2(3) + 3 − 9 = 0. The gradient of AB is 1/2 and the locus has gradient −2, so the line is perpendicular to AB, as expected.
Worked example 2: a distance ratio
P moves so that PA : PB = 1 : 2, where A(0, 0) and B(6, 0).
The rule PA : PB = 1 : 2 means 2PA = PB. Square both sides: 4PA² = PB².
So 4(x² + y²) = (x − 6)² + y². Expand: 4x² + 4y² = x² − 12x + 36 + y², which gives 3x² + 3y² + 12x − 36 = 0, then x² + y² + 4x − 12 = 0.
Completing the square gives (x + 2)² + y² = 16, a circle with centre (−2, 0) and radius 4.
Check with P(2, 0): PA = 2 and PB = 4, so the ratio is 1 : 2. Also P(−6, 0): PA = 6, PB = 12. Both satisfy the rule.
The mistake that costs marks
The ratio locus is where most students slip, because the ratio can be placed on the wrong side.
| Step | Wrong | Right |
|---|---|---|
| Read PA : PB = 1 : 2 | PA = 2PB | 2PA = PB |
| Square | 4PA = PB² | 4PA² = PB² |
| Test P(2, 0), which lies on the locus | PA = 2PB gives 2 = 8, false | 2PA = PB gives 4 = 4, true |
A check point exposes a reversed ratio at once, because the wrong rule fails for a point you know is on the locus.
Check yourself
P moves so that PA : PB = 1 : 3, where A(0, 0) and B(8, 0). Find the equation of the locus.
Answer
3PA = PB, so 9PA² = PB².
9(x² + y²) = (x − 8)² + y². Expand: 9x² + 9y² = x² − 16x + 64 + y².
So 8x² + 8y² + 16x − 64 = 0, which simplifies to x² + y² + 2x − 8 = 0.
Check with P(2, 0): PA = 2, PB = 6, ratio 1 : 3. Also (x + 1)² + y² = 9, a circle with centre (−1, 0) and radius 3.
What to study next
Test the whole chapter with the coordinate geometry practice set. For mixed problems where the approach is not obvious, continue with geometry problems with several possible approaches.
To have a teacher set you fresh locus rules and watch your working, see online one-to-one Additional Mathematics tuition.