Coordinate Geometry

Coordinate Geometry: key points

• Gradient between (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁) / (x₂ − x₁).
• Equation of a straight line: y − y₁ = m(x − x₁), or y = mx + c.
• Parallel lines have equal gradients; perpendicular gradients multiply to −1.
• Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2); distance = √((x₂ − x₁)² + (y₂ − y₁)²).
• Circle with centre (a, b) and radius r: (x − a)² + (y − b)² = r².
• Complete the square to find the centre and radius from x² + y² + 2fx + 2gy + c = 0.
• A tangent is perpendicular to the radius at the point of contact.
• The perpendicular bisector of a chord passes through the centre of the circle.