Write circle equations and identify their centres
Move between centre-radius form and the expanded equation by completing the square.
Section 8 · Coordinate geometry of the circle
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which equation represents a circle with centre (2, −3) and radius 4?
What you will learn
- Use the equation of a circle with a stated centre and radius
- Identify centre and radius from standard form
- Complete squares to recover centre and radius from general form
Centre-radius form makes the geometry visible
A circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r². The centre is (a, b), and r is positive.
A point lies on the circle when its coordinates make the left side equal r². A value below r² is inside; a value above r² is outside.
Complete the square in the general equation
For x² + y² + 2gx + 2fy + c = 0, complete each coordinate square: (x + g)² + (y + f)² = g² + f² − c.
The centre is (−g, −f), and the radius is √(g² + f² − c). A real circle requires the squared radius to be positive; zero gives a single point and a negative value gives no real points.
Keep the sign inside each bracket: x² − 6x completes to (x − 3)² − 9, while y² + 4y completes to (y + 2)² − 4.
Pause and explain
What is the centre of x² + y² + 8x − 10y + 25 = 0?
Worked example
Find the centre and radius of x² + y² − 6x + 4y − 12 = 0.
Show the worked solution
- Group x and y terms and move the constant: x² − 6x + y² + 4y = 12.
- Complete squares: (x − 3)² − 9 + (y + 2)² − 4 = 12.
- Thus (x − 3)² + (y + 2)² = 25.
Answer Centre (3, −2); radius 5.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the centre and radius of (x + 1)² + (y − 4)² = 36.
Give me a hint
Compare each bracket with (x − a)² and read r².
Compare my reasoning
- The x bracket gives a = −1.
- The y bracket gives b = 4.
- r² = 36, so r = 6.
Centre (−1, 4); radius 6.
Look for these in your work
- Centre signs interpreted
- Both coordinates identified
- Radius square-rooted
Complete the square and find the centre and radius of x² + y² + 8x − 10y + 25 = 0.
Give me a hint
Group x and y terms, then balance both completed squares.
Compare my reasoning
- x² + 8x = (x + 4)² − 16; y² − 10y = (y − 5)² − 25.
- The equation becomes (x + 4)² + (y − 5)² = 16.
- Read centre (−4, 5) and radius 4.
Centre (−4, 5); radius 4.
Look for these in your work
- Squares completed accurately
- Standard form reached
- Centre and radius identified
A circular garden is centred at (3, −2) and has radius 5 m. Write its equation and decide whether the point (6, 2) lies on its boundary.
Give me a hint
Use centre-radius form and compare the squared distance with r².
Compare my reasoning
- The equation is (x − 3)² + (y + 2)² = 25.
- At (6, 2), the left side is 3² + 4² = 25.
- The point is exactly 5 m from the centre, so it lies on the boundary.
(x − 3)² + (y + 2)² = 25; (6, 2) lies on the circle.
Look for these in your work
- Equation written
- Squared distance evaluated
- Boundary classification justified
Common mistakes
- Reversing the sign of the centre coordinates
- Forgetting to balance the constants when completing squares
- Calling r² the radius instead of taking its square root
What can you explain now?
What are the centre and radius of (x + 1)² + (y − 4)² = 36?
Compare with the explanation
Centre (−1, 4), radius 6
Read the centre using the opposite signs inside the brackets, then take √36.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do you read the centre and radius from centre-radius form, and recover them from general form?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.