Coordinate geometry
Coordinate geometry tests straight lines and circles in the Cartesian plane: equations, gradients, distances, intersections and tangency. Questions typically combine a circle given in expanded form with a line, asking for the centre, radius, points of intersection or a tangent length.
Straight lines
A line through (x₁, y₁) with gradient m has equation y − y₁ = m(x − x₁). Parallel lines share a gradient; perpendicular gradients satisfy m₁m₂ = −1. The distance between two points is √((x₂ − x₁)² + (y₂ − y₁)²), and the midpoint is the average of the coordinates.
Circles
A circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r². When given in expanded form x² + y² + 2fx + 2gy + c = 0, complete the square in both variables: the centre is (−f, −g) and the radius is √(f² + g² − c). The perpendicular from the centre to a chord bisects it, and a tangent is perpendicular to the radius at the point of contact.
Intersections and tangency
Substitute the line into the circle to obtain a quadratic in one variable. Two distinct roots mean two intersection points, a repeated root means tangency, and no real roots means the line misses the circle. The discriminant therefore decides the geometric relationship.
Worked example
Find the centre and radius of the circle x² + y² − 6x + 8y − 11 = 0.
Complete the square: (x² − 6x + 9) + (y² + 8y + 16) = 11 + 9 + 16 = 36. So (x − 3)² + (y + 4)² = 36, giving centre (3, −4) and radius 6.
Traps that cost marks
- Reading the radius as r rather than r² from the right-hand side.
- Sign errors when completing the square, especially with a negative y coefficient.
- Forgetting to add the completed-square constants to both sides.
- Assuming a line is a tangent without verifying the discriminant is zero.
- Using the gradient of the radius instead of its negative reciprocal for the tangent.