6.2PureCore

Equation of a Circle

A circle is the set of points a fixed distance (the radius) from a centre. That single idea, combined with Pythagoras, gives the equation of any circle — and lets you read off its centre and radius at a glance.

30 min Video by Zeeshan Zamurred Circles

Edexcel AS Level Maths: 6.2 Equation of a CircleWatch the full walkthrough before the notes below.

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What you'll be able to do

Write the equation of a circle from its centre and radius
Read the centre and radius from a circle equation
Find the centre and radius by completing the square
Determine whether a point lies on, inside or outside a circle

The standard form

A circle with centre $(a, b)$ and radius $r$ has a beautifully simple equation, derived straight from Pythagoras.

(x - a)^{2} + (y - b)^{2} = r^{2}

Centre

(a, b)

, radius

r

. Watch the signs: the centre coordinates are subtracted.

1Substitute

a = 2

b = - 3

r = 5

2Note

- (- 3) = + 3

inside the bracket.

Answer

(x - 2)^{2} + (y + 3)^{2} = 25

Tip — The centre coordinates appear with the OPPOSITE sign inside the brackets: (x − a) means centre x = a.

Reading off centre and radius

Given the standard form, the centre is read directly (with sign flips) and the radius is the square root of the right-hand side.

1Centre: flip signs →

(- 1, 4)

2Radius:

9 = 3

Answercentre

(- 1, 4)

, radius

3

Completing the square form

Circles are often given expanded, like $x^{2} + y^{2} + 2 x - 6 y - 6 = 0$ . Complete the square on the $x$ terms and on the $y$ terms to recover the standard form.

1Group:

(x^{2} + 2 x) + (y^{2} - 6 y) = 6

2Complete the square:

(x + 1)^{2} - 1 + (y - 3)^{2} - 9 = 6

3Tidy:

(x + 1)^{2} + (y - 3)^{2} = 16

Answercentre

(- 1, 3)

, radius

4

Tip — Move the constant to the right and remember to add back the numbers created by completing each square.

Formula recap

(x - a)^{2} + (y - b)^{2} = r^{2}

Centre (a, b), radius r.

centre = (a, b), r = RHS

Reading the circle.

Common mistakes to avoid

Reading the centre of (x + 1)² + (y − 4)² = 9 as (1, −4).

Flip the signs: the centre is (−1, 4).

Taking the radius as the right-hand side instead of its square root.

r² = 9 means r = 3, not 9.

Key takeaways

Standard form: (x − a)² + (y − b)² = r².
Centre coordinates have opposite signs to the brackets; radius = √(RHS).
For an expanded equation, complete the square in x and y to find centre and radius.

Test yourself

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