amazon.co.ukCoordinate Geometry
A-Level Mathematics revision: Coordinate Geometry. Learning objectives, key points, worked examples and practice questions across AQA, Edexcel, OCR, WJEC and CCEA.
📌 Key Points
Key Fact: Gradient m = (y₂ - y₁)/(x₂ - x₁)
Key Fact: Line equations: y - y₁ = m(x - x₁) or y = mx + c
Key Fact: Distance d = sqrt((x₂ - x₁)^2 + (y₂ - y₁)^2)
Key Fact: Midpoint M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Key Fact: Parallel lines: equal gradients (m₁ = m₂)
Key Fact: Perpendicular lines: product of gradients = -1 (m₁ x m₂ = -1)
Key Fact: Circle: (x - a)^2 + (y - b)^2 = r^2, centre (a,b), radius r
Key Fact: Circle properties: radius ⟂ tangent, perpendicular bisector of chord passes through centre
Key Fact: Intersection: solve simultaneous equations of line and circle/line
🎯 Learning Objectives
- Find the equation of a line given gradient and point, or two points
- Compute distance between two points and midpoint of a segment
- Determine parallel and perpendicular lines using gradients
- Find intersection of two lines algebraically
- Understand equation of a circle (x-a)^2 + (y-b)^2 = r^2
- Find tangents and normals to circles and curves
- Apply coordinate geometry to solve geometric problems
💡 Worked Example
Exam-Style Question
Question: Find the equation of the circle with diameter endpoints A(1,2) and B(5,6). Find the tangent at A.
Model Answer:
Centre = midpoint of AB = (3,4). Radius = half of AB = ½sqrt((5-1)^2+(6-2)^2) = ½sqrt32 = 2sqrt2. Circle: (x-3)^2+(y-4)^2=8. Gradient of radius OA = (2-4)/(1-3)=1. Tangent gradient = -1. Tangent at A: y-2 = -1(x-1) -> y = -x+3
❓ Practice Questions
Model answers are being added progressively - questions marked ✗ don't have one yet. Cross-check with your teacher or the official mark scheme.
Questions:
- Find the line through (-2,3) perpendicular to 2x + 3y = 6✗ answer coming soon
- Find intersection of y = 2x - 1 and 3x + y = 7✗ answer coming soon
- Circle C: (x-2)^2 + (y+1)^2 = 25. Find tangent at (5,3)✗ answer coming soon
- Points A(1,1), B(4,5), C(6,2). Show triangle ABC is right-angled✗ answer coming soon
- Find the locus of points equidistant from (2,3) and the line y = -1✗ answer coming soon
🎬 Video Resources
- Coordinate Geometry - Overview -- YouTube Search
- Coordinate Geometry - Exam Questions -- YouTube Search
📄 Past Papers & Exam Resources
🔗 Further Reading & Resources
- Physics & Maths Tutor - Coordinate Geometry -- Free notes, worksheets, and past paper questions by topic
- Dr Frost Maths - Coordinate Geometry -- Free resources, slides, and interactive questions
- BBC Bitesize - Coordinate Geometry -- BBC Bitesize revision resources
📚 Lesson Plan (50 minutes)
- Starter (5 min): Recall prior knowledge of coordinate geometry with quick questions.
- Teaching (15 min): Work through each of the learning objectives, explaining principles step by step.
- Key points review (5 min): Revisit the key points together, confirming understanding.
- Worked example (10 min): Model the example question: Find the equation of the circle with diameter endpoints A(1,2) and B(5,6). Find the tangent at A.. Solution: Centre = midpoint of AB = (3,4). Radius = half of AB = ½sqrt((5-1)^2+(6-2)^2) = ½sqrt32 = 2sqrt2. Circle: (x-3)^2+(y-4)^2=8. Gradient of radius OA = (2-4)/(1-3)=1. Tangent gradient = -1. Tangent at A: y-2 = -1(x-1) -> y = -x+3
- Practice (10 min): Students attempt the practice questions independently; circulate and support.
- Plenary (5 min): Review answers and address misconceptions.
🏠 Homework
- Find the line through (-2,3) perpendicular to 2x + 3y = 6
- Find intersection of y = 2x - 1 and 3x + y = 7
- Circle C: (x-2)^2 + (y+1)^2 = 25. Find tangent at (5,3)
- Points A(1,1), B(4,5), C(6,2). Show triangle ABC is right-angled
- Find the locus of points equidistant from (2,3) and the line y = -1
🧾 Assessment
Check practice answers against the model answer; use the built-in practice questions as formative assessment.