What is the centre of the circle (x − 2)² + (y + 5)² = 16?

(2, −5). The signs are the opposite of those in the brackets.

Which is the equation of the circle with centre (0, 3) and radius 2?

x² + (y − 3)² = 4. (x − 0)² + (y − 3)² = 2².

What is the centre of x² + y² − 4x + 10y = 0?

(2, −5). Halve the coefficients of x and y and change the signs.

The equation of a circle · Cambridge IGCSE Additional Mathematics

In plain words

A circle is every point at the same distance from its centre. Write that sentence in coordinates, using Pythagoras, and you have its equation.

5 things to know

A circle with centre (a, b) and radius r has the equation (x − a)² + (y − b)² = r². This is on the formula list.
Read the centre with the signs changed: (x − 3)² + (y + 2)² = 25 has centre (3, −2) and radius 5.
The expanded form is x² + y² + 2gx + 2fy + c = 0. Its centre is (−g, −f) and its radius is √(g² + f² − c).
To find the centre and radius from the expanded form, complete the square for x and for y.
To write the equation of a circle, you need its centre and its radius. The radius is the distance from the centre to any point on the circle.

Worked example

Find the centre and radius of the circle x² + y² − 6x + 4y − 12 = 0.

Complete the square: (x − 3)² − 9 + (y + 2)² − 4 − 12 = 0.
(x − 3)² + (y + 2)² = 25.
The centre is (3, −2) and the radius is 5.

Worked example

Find the equation of the circle with centre (2, −1) that passes through (5, 3).

r² = (5 − 2)² + (3 − (−1))² = 9 + 16 = 25.
(x − 2)² + (y + 1)² = 25.

Tips and tricks

The right-hand side is r², not r. A circle with "= 25" has radius 5.
The ends of a diameter give the centre (their midpoint) and the radius (half their distance apart).

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

The equation of a circle: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What is the centre of the circle (x − 2)² + (y + 5)² = 16?
- (−2, 5)
- (2, −5) (the answer)
- (2, 5)
- (−2, −5)
The signs are the opposite of those in the brackets.
What is the radius of the circle x² + y² = 36?
- 36
- 18
- 6 (the answer)
- 12
r² = 36.
Which is the equation of the circle with centre (0, 3) and radius 2?
- x² + (y + 3)² = 4
- x² + (y − 3)² = 4 (the answer)
- x² + (y − 3)² = 2
- (x − 3)² + y² = 4
(x − 0)² + (y − 3)² = 2².
A circle has centre (1, 1) and passes through (4, 5). What is its radius?
- 3
- 4
- 5 (the answer)
- 7
√(3² + 4²).
What is the centre of x² + y² − 4x + 10y = 0?
- (−2, 5)
- (2, −5) (the answer)
- (4, −10)
- (−4, 10)
Halve the coefficients of x and y and change the signs.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 8 marks. Write your answers on paper, then check them.