Find the equation of a locus
A standard method from Loci: the steps and worked examples.
Steps
- Let the moving point be P(x, y).
- Write the condition in words, then with the distance formula.
- Square both sides to remove the roots.
- Expand and simplify.
- Name it ("the locus is a straight line / circle / parabola") and describe it if asked.
Worked examples
Example 1: P is equidistant from A(1, 2) and B(5, 6). Find the equation of the locus of P and describe the geometric relationship.
PA = PB
(x − 1)² + (y − 2)² = (x − 5)² + (y − 6)²
−2x + 1 − 4y + 4 = −10x + 25 − 12y + 36
8x + 8y − 56 = 0
x + y − 7 = 0Check: the mid-point of AB, (3, 4), satisfies it, and the slope −1 is perpendicular to AB's slope 1. It is the perpendicular bisector of AB.
Example 2: P is equidistant from the point (0, 1) and the line y = −1. Find the locus.
√[x² + (y − 1)²] = |y + 1|
x² + y² − 2y + 1 = y² + 2y + 1
x² = 4y, i.e. y = x²/4: a parabolaFor the whole topic, read Loci: key points, common mistakes and how the DSE tests it.