Equations of circles
A circle is every point at the same distance from its centre, and Pythagoras turns that into an equation. Circle questions combine it with straight-line work and the GCSE circle theorems.
Part 1 of 3: Learn it
In short
- (x − a)² + (y − b)² = r² has centre (a, b) and radius r.
- From the expanded form, complete the square in x and in y.
- The radius drawn to the point of contact meets the tangent at a right angle.
Where this is in your specification
Spec points: DfE C2 (AQA 7357 C2, Edexcel 9MA0 Pure topic 3, OCR A H240 1.03)
| Board | Topic: Coordinate geometry and parametric equations |
|---|---|
| DfE content | C1-C4 |
| AQA 7357 | C1-C4 |
| Edexcel 9MA0 | Pure topic 3 |
| OCR H240 | 1.03 |
| Higher C847 76 | Geometric skills: the straight line, the circle |
The equation
Watch the signs: (x + 2)² + (y − 5)² = 36 has centre (−2, 5) and radius 6. A circle with centre at the origin is x² + y² = r².
Completing the square
Group the x terms and the y terms, complete the square for each, and move the constants to the right-hand side. If the right-hand side comes out as zero or negative, the equation does not describe a real circle.
Geometry facts to use
- The tangent at a point is perpendicular to the radius to that point.
- The perpendicular bisector of any chord passes through the centre.
- An angle in a semicircle is a right angle.
- A point is inside the circle if its distance from the centre is less than r.
Write down the centre and radius of (x − 6)² + (y + 1)² = 20.
Show the answer
The centre is (6, −1) and the radius is √20 = 2√5.
Part 2 of 3: See it worked
Worked examples
Example 1
Find the centre and radius of x² + y² − 6x + 4y − 12 = 0.
- x² − 6x = (x − 3)² − 9 and y² + 4y = (y + 2)² − 4
- (x − 3)² − 9 + (y + 2)² − 4 − 12 = 0
- (x − 3)² + (y + 2)² = 25
Answer: Centre (3, −2), radius 5.
Example 2
Find the equation of the tangent to (x − 3)² + (y + 2)² = 25 at the point (6, 2).
- Gradient of the radius from (3, −2) to (6, 2): 4 ÷ 3
- Tangent gradient: −3/4
- y − 2 = −¾(x − 6), so 4y − 8 = −3x + 18
Answer: 3x + 4y = 26.
Common mistakes
- Getting the sign of the centre wrong: (x + 2)² means a = −2.
- Giving r² as the radius.
- Forgetting to move the constants from completing the square to the other side.
- Using the gradient of the radius as the gradient of the tangent.
Is the point (2, 1) inside, on or outside x² + y² = 9?
Show the answer
Inside: 2² + 1² = 5, which is less than 9.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Write down the centre and radius of (x − 6)² + (y + 1)² = 20.
The centre is (6, −1) and the radius is √20 = 2√5.
Is the point (2, 1) inside, on or outside x² + y² = 9?
Inside: 2² + 1² = 5, which is less than 9.
Which line through a circle always passes through the centre: the perpendicular bisector of a chord, or the tangent?
The perpendicular bisector of a chord.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses