Coordinate Geometry
Core ideas
Candidates must handle straight lines (gradient, midpoint, length, equation in various forms), parallel and perpendicular lines, and circles. The gradient between two points is (y₂−y₁)/(x₂−x₁); perpendicular gradients satisfy m₁m₂ = −1. The circle with centre (a, b) and radius r has equation (x−a)² + (y−b)² = r²; expanding gives x² + y² + 2gx + 2fy + c = 0, with centre (−g, −f) and radius √(g² + f² − c). Also needed: finding intersections of a line and a circle, and using the discriminant to decide whether a line is a tangent, chord, or misses the circle.
Traps
- Sign errors reading the centre from (x−a)² + (y−b)².
- Forgetting to complete the square when the circle equation is expanded.
- Using the negative reciprocal the wrong way round for perpendicular lines.
- Mixing up midpoint and gradient formulas.
- Assuming a line meets a circle twice without checking the discriminant.
Worked example
Find the centre and radius of x² + y² − 6x + 4y − 12 = 0. Complete the square: (x−3)² − 9 + (y+2)² − 4 − 12 = 0, so (x−3)² + (y+2)² = 25. Centre (3, −2), radius 5.
Rule of thumb
Convert any circle equation to centre-radius form before answering; the centre is the opposite sign of the linear coefficients.