Coordinate Geometry
Coordinate Geometry
Coordinate geometry in this paper covers straight lines, circles and the interplay between them: gradients, perpendicularity, distance, midpoints, the equation of a circle in both expanded and completed-square form, tangents and normals, and intersections of a line with a circle. Questions are often multi-part, building from an equation to a geometric property and then to an area.
Straight lines
Gradient m = (y₂ − y₁)/(x₂ − x₁). Forms to keep ready: y − y₁ = m(x − x₁), y = mx + c, and ax + by + c = 0 whose gradient is −a/b. Perpendicular lines satisfy m₁m₂ = −1. Distance between points is √((x₂ − x₁)² + (y₂ − y₁)²) and the midpoint is the average of the coordinates.
Circles
Centre (a, b), radius r: (x − a)² + (y − b)² = r². Given x² + y² + 2gx + 2fy + c = 0, complete the square: centre (−g, −f) and radius √(g² + f² − c). The tangent at a point is perpendicular to the radius there, so its gradient is the negative reciprocal of the radius gradient; the normal passes through the centre.
Line meets circle
Substitute the line into the circle to get a quadratic. Two distinct roots: two intersection points (secant). One repeated root: tangent, so the discriminant is zero. No real roots: the line misses the circle. The perpendicular distance from centre to line, |ax₀ + by₀ + c|/√(a² + b²), compared with r, gives the same information faster.
Worked example
Find the centre and radius of x² + y² − 6x + 4y − 12 = 0 and determine whether y = 2x − 1 cuts it. Completing squares: (x − 3)² − 9 + (y + 2)² − 4 − 12 = 0, so (x − 3)² + (y + 2)² = 25. Centre (3, −2), radius 5. Rewrite the line as 2x − y − 1 = 0. Distance from centre: |2(3) − (−2) − 1|/√(4 + 1) = 7/√5 ≈ 3.13. Since 3.13 < 5, the line cuts the circle in two points.
Traps that cost marks
- Sign errors when completing the square, giving the centre as (3, 2) instead of (3, −2).
- Using the radius gradient rather than its negative reciprocal for a tangent.
- Forgetting the radius is √(g² + f² − c), not g² + f² − c.
- Leaving an answer as a decimal where the exact surd is required.