Coordinate Geometry
Coordinate Geometry
This domain covers straight lines, circles and the interplay between them: gradients, distances, perpendicularity, the equation of a circle in both forms, and tangency decided by the discriminant. Questions often chain - find the centre, then the tangent, then where it meets an axis.
Core rules
- Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).
- Gradient m = (y₂ − y₁)/(x₂ − x₁); parallel lines share m; perpendicular lines satisfy m₁m₂ = −1.
- Line forms: y − y₁ = m(x − x₁); the line ax + by + c = 0 has gradient −a/b.
- Circle: (x − a)² + (y − b)² = r² has centre (a, b) and radius r. The expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and r² = g² + f² − c.
- A tangent is perpendicular to the radius at the point of contact; a line meets a circle in two, one or no points according as the discriminant is positive, zero or negative.
Worked example
Find the centre and radius of x² + y² − 6x + 4y − 12 = 0. Complete the square: (x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4, so (x − 3)² + (y + 2)² = 25. The centre is (3, −2) and the radius is 5. The commonest error is leaving 12 on the right - whatever you add inside the brackets must also be added to the constant side.
Traps
- Writing r in place of r² in the circle equation.
- Sign errors in the centre: (y + 2)² gives b = −2.
- Using the radius gradient rather than its negative reciprocal for a tangent.
- Forgetting that the perpendicular bisector of a chord passes through the centre, often the quickest route.