Coordinate Geometry
Core concepts
Candidates must work confidently with straight lines (gradient, midpoint, distance, parallel and perpendicular conditions), circles in both expanded and completed-square form, and the intersection of lines and curves. Know that a tangent meets a circle at exactly one point, and be able to use the discriminant or perpendicular-distance argument to establish tangency.
Traps
- The perpendicular gradient is the negative reciprocal; a common error is forgetting the sign change.
- For a circle x² + y² + 2gx + 2fy + c = 0, the centre is (−g, −f) and the radius is √(g² + f² − c). Sign slips here are frequent.
- When finding intersections by substitution, always report the coordinates, not just the x-values.
- The perpendicular distance from a point to a line uses |ax₀ + by₀ + c| / √(a² + b²); forgetting the modulus loses a case.
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: half the x-coefficient with sign reversed gives the centre's x-coordinate; the same for y.