"Describe the geometric relationship"
A standard method from Loci: the steps and worked examples.
Steps
Say what the locus is and how it relates to the given objects: "the perpendicular bisector of AB", not just "a straight line".
- Name the shape: a straight line, a circle or a parabola.
- Name how it relates to the given points or lines: "the perpendicular bisector of AB", "the circle with centre C and radius 3".
- Check with one point: the mid-point of AB, or the centre, should fit the equation or the description.
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.
For the whole topic, read Loci: key points, common mistakes and how the DSE tests it.