Coordinate geometry
What this domain covers
Coordinate geometry in 8227 covers straight lines (gradient, midpoint, distance, equations), perpendicularity, circle equations in both centre–radius and expanded form, tangents and normals to circles, intersections of lines with curves, and curve sketching from intercepts, asymptotes and turning points.
Core rules
- Gradient m = (y₂ − y₁)/(x₂ − x₁); parallel lines share the same m; perpendicular gradients satisfy m₁m₂ = −1.
- Line forms: y − y₁ = m(x − x₁), and ax + by + c = 0 in general form.
- Distance between points: √((x₂ − x₁)² + (y₂ − y₁)²); midpoint is the average of the coordinates.
- Circle: (x − a)² + (y − b)² = r² has centre (a, b) and radius r. The expanded form x² + y² + 2gx + 2fy + c = 0 is converted by completing the square in both variables.
- A tangent at P is perpendicular to the radius CP; the perpendicular from the centre to a chord bisects that chord.
- A line meets a circle in 0, 1 or 2 points according to the discriminant of the quadratic obtained by substitution.
Worked example
Find the centre and radius of x² + y² − 6x + 4y − 12 = 0. Group the terms: (x² − 6x) + (y² + 4y) = 12. Completing both squares gives (x − 3)² − 9 + (y + 2)² − 4 = 12, so (x − 3)² + (y + 2)² = 25. The centre is (3, −2) and the radius is 5. Substituting the centre into the original left-hand side gives −25, which confirms r² = 25.
Traps
- Reading the centre as (−3, 2) from (x − 3)² + (y + 2)²; the signs reverse.
- Treating r² as the radius instead of taking the square root.
- Using m rather than −1/m for a normal, or the radius gradient rather than the tangent gradient.
- Forgetting that an expanded circle equation yields no real locus when g² + f² − c < 0.