CirclesExam Questions
A-level Coordinate Geometry · Edexcel, AQA and OCR · Builds on: straight lines and completing the square
The equation of a circle
Standard form
A circle with centre (a, b) and radius r has equation (x – a)² + (y – b)² = r². Read it straight off: the numbers inside the brackets give the centre (watch the signs), the number on the right is the radius squared.
Some boards also ask for the expanded form x² + y² + ax + by + c = 0 – multiply out the brackets and collect. To go back the other way, complete the square on the x-terms and the y-terms.
Every circle question comes down to two facts: the centre and the radius. Given the centre and one point on the circle, the radius is the distance between them – and since the equation only needs r², you can usually leave it as a whole number or surd.
Equation of a circle: 5 steps
- Write down the standard form (x – a)² + (y – b)² = r² so you can see what you need: a centre (a, b) and a radius squared.
- Pin down the centre. It is usually given. If you are given a diameter instead, the centre is the midpoint of the two endpoints.
- Find r² from a point on the circle. Square the distance from the centre to the given point: r² = (x₁ – a)² + (y₁ – b)². No square root needed – the equation wants r².
- Assemble the equation and, if the question asks for it, multiply out into the form x² + y² + ax + by + c = 0.
- Check with the given point. Substitute it back into your equation – both sides must balance.
Worked examples
Example 1 – centre and radius given
A circle has centre (3, –2) and radius 5.
Straight into the standard form: (x – 3)² + (y + 2)² = 25. Note the +2 – the y-coordinate of the centre is negative, so the bracket flips sign.
Example 2 – centre and a point on the circle
A circle has centre (–1, 7) and passes through (0, 0).
r² = (0 – (–1))² + (0 – 7)² = 1 + 49 = 50, so the equation is (x + 1)² + (y – 7)² = 50. Check: substituting (0, 0) gives 1 + 49 = 50. It balances.
The mistake examiners see most
Sign slips and the missing square root
A centre of (–1, 7) gives brackets (x + 1) and (y – 7) – the signs inside the brackets are the OPPOSITE of the centre coordinates. Candidates regularly write (x – 1)² for a centre at x = –1.
The other classic: computing r² = 50 and then writing = 50² or taking a square root you do not need. The equation wants r² exactly as you found it.
Examiner tip
Say the centre and radius out loud
The method mark is for using (x – a)² + (y – b)² = r² correctly, so write the centre and r² down explicitly before assembling the equation. If the question asks for the expanded form, you only get the final mark if every bracket is multiplied out and collected – and if it asks for a tangent, remember the radius to the point of contact is perpendicular to the tangent, so its gradient gives you the tangent gradient instantly.
Exam questions with video solutions
Try each question yourself first, then open the full video solution to check your working. Question 1 is from Edexcel C2 January 2012; Question 2 is from OCR C1 January 2012. The topics these questions draw on are linked under Helpful Tutorials.
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