Equation of a circle: centre, radius and tangents
A circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r². At GCSE you meet circles centred on the origin, x² + y² = r²; at A level, any centre. This guide covers both, with tangents, completing the square and intersections.
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The equation of a circle
Every point (x, y) on a circle is the same distance r from the centre. By Pythagoras, the distance from (a, b) to (x, y) is √((x − a)² + (y − b)²). Setting that equal to r and squaring gives:
(x − a)² + (y − b)² = r², a circle with centre (a, b) and radius r.
When the centre is the origin, a = b = 0 and the equation is x² + y² = r². That is the form used at GCSE Higher tier.
| Equation | Centre | Radius |
|---|---|---|
| x² + y² = 49 | (0, 0) | 7 |
| (x − 2)² + (y + 5)² = 16 | (2, −5) | 4 |
| (x + 1)² + y² = 10 | (−1, 0) | √10 |
| x² + y² − 6x + 4y − 12 = 0 | (3, −2) | 5 (see Example 2) |
Watch the signs: (y + 5)² means b = −5, because y + 5 = y − (−5).
Worked examples
Example 1: writing the equation
Write the equation of the circle with centre (2, −5) and radius 4.
- Substitute a = 2, b = −5, r = 4.
- (x − 2)² + (y + 5)² = 16. Remember to square the radius.
Example 2: completing the square (A level)
Find the centre and radius of x² + y² − 6x + 4y − 12 = 0.
- Group the x terms and the y terms: (x² − 6x) + (y² + 4y) = 12.
- Complete the square: (x − 3)² − 9 + (y + 2)² − 4 = 12.
- (x − 3)² + (y + 2)² = 25.
- Centre (3, −2), radius 5.
Example 3: a tangent (GCSE Higher)
Find the equation of the tangent to x² + y² = 25 at the point (3, 4).
- A tangent is perpendicular to the radius at the point where it touches.
- The radius from (0, 0) to (3, 4) has gradient 4/3, so the tangent has gradient −3/4.
- y − 4 = −(3/4)(x − 3), which simplifies to 3x + 4y = 25.
Example 4: where a line meets a circle
Find where the line y = x + 1 meets the circle x² + y² = 13.
- Substitute: x² + (x + 1)² = 13, so 2x² + 2x − 12 = 0, or x² + x − 6 = 0.
- (x + 3)(x − 2) = 0, so x = 2 or x = −3.
- The points are (2, 3) and (−3, −2). Check: 4 + 9 = 13 and 9 + 4 = 13.
Example 5: a circle from a diameter
A and B are (1, 2) and (7, 10). Find the equation of the circle with diameter AB.
- The centre is the midpoint of AB: (4, 6).
- AB = √(6² + 8²) = 10, so the radius is 5.
- (x − 4)² + (y − 6)² = 25.
Inside, on or outside the circle?
Substitute the point into the left-hand side and compare with r². For x² + y² = 9 and the point (1, 2): 1 + 4 = 5, which is less than 9, so the point is inside the circle. Equal to r² means on the circle; greater means outside.
How many times does a line meet a circle?
After substituting the line into the circle, you get a quadratic. Its discriminant b² − 4ac tells you: positive means two points, zero means one (the line is a tangent), and negative means the line misses the circle. Our quadratic formula guide explains the discriminant.
For circle theorems, such as the angle in a semicircle and the tangent-radius rule, see our circle theorems guide.
Practice questions
Try each one before opening the answer.
Q1. Write the equation of the circle with centre (2, −5) and radius 4.
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Q2. What is the radius of x² + y² = 49?
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Q3. Find the centre and radius of x² + y² + 8x − 2y + 8 = 0.
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Q4. Find the equation of the tangent to x² + y² = 10 at (1, 3).
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Q5. Does the point (5, 1) lie on (x − 2)² + (y + 3)² = 25?
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Common mistakes
- Forgetting to square the radius, or to square-root r² when asked for the radius.
- Sign errors: (x + 3)² means the centre has x = −3.
- Using the gradient of the radius for the tangent, instead of its negative reciprocal.
- Dropping the constants when completing the square.
Frequently asked questions
What is the equation of a circle?
(x − a)² + (y − b)² = r², for centre (a, b) and radius r. With centre at the origin it is x² + y² = r².
How do you find the centre and radius from an expanded equation?
Complete the square in x and in y, then read the centre and radius from (x − a)² + (y − b)² = r².
Is the equation of a circle on GCSE Maths?
Yes, at Higher tier, for circles centred on the origin, including the tangent at a point. Circles with any centre are in A level Maths.
How do you find a tangent to a circle?
Find the gradient of the radius to the point, take its negative reciprocal, and use y − y₁ = m(x − x₁).
Key takeaways
- (x − a)² + (y − b)² = r²: centre (a, b), radius r.
- Complete the square to find the centre and radius from an expanded equation.
- A tangent is perpendicular to the radius at the point of contact.