A-level · Coordinate geometry

Equations of a circle

Connect centre and radius with expanded equations, tangents and line intersections.

Drag C to move the centre; drag R sideways to change the radius.

(x−2)2+(y+1)2=9(x-2)^2+(y+1)^2=9
Coordinate graph. Both axes use the same scale. Coloured rings are draggable handles.-12-12-8-8-4-444881212xy0CR

Tab to a handle, then use the arrow keys. Use the sliders for precise values.

Every point on the circle is 3 units from C (2, -1).

Centre
C=(2,−1)C=(2,-1)
Radius
r=3,r2=9r=3,\quad r^2=9
Centre and radius form
(x−2)2+(y+1)2=9(x-2)^2+(y+1)^2=9
Expanded form
x2+y2−4x+2y−4=0x^2+y^2-4x+2y-4=0

The centre makes both brackets zero. The right-hand side is the radius squared, not the radius.

Predict, test and explain

Step 1 of 3

Predict

For (x − 2)² + (y + 1)² = 9, which centre and radius should the diagram show?

InvestigateMove the centre without changing the radius. Which coefficients change?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Find the centre and radius of this circle.

x2+y2−6x+4y+4=0x^2+y^2-6x+4y+4=0

Hint

Complete the squares in x and y separately, then compare with (x − h)² + (y − k)² = r².

Reveal answer

The completed form has squared displacements from (3, −2) and right-hand side 9. The radius is the positive square root of 9.

(x−3)2+(y+2)2=9⇒C=(3,−2), r=3(x-3)^2+(y+2)^2=9\quad\Rightarrow\quad C=(3,-2),\ r=3

Build an explanation

Hint 1

The centre is the point that makes both squared brackets zero.

Hint 2

The right-hand side is the radius squared. Take its positive square root to find the radius.

Hint 3

For an expanded equation, complete the square in x and y separately. A tangent is perpendicular to the radius at its point of contact.

Worked example

This example uses fixed values, separate from the diagram controls.

Find the centre and radius of x² + y² − 4x + 2y − 4 = 0.

  1. Rearrange to x² − 4x + y² + 2y = 4.
  2. Complete the squares: (x − 2)² − 4 + (y + 1)² − 1 = 4.
  3. Therefore (x − 2)² + (y + 1)² = 9.

Answer: Centre (2, −1); radius 3.

Connect this idea

Watch out: Read the centre by solving each bracket equal to zero. In (x − 2)² + (y + 1)² = 9, the centre is (2, −1) and the radius is 3, not 9.

Specification and learning route

AQA C2 · Edexcel Pure 3.2

AS and A-level. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Coordinates and distance; Pythagoras; Completing the square; Straight-line gradients for tangents.

AQA specification · Edexcel specification

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