Circles

Equation of a Circle

The equation of a circle which is centred at the origin and has radius a is:

• x² + y² = a²

The equation of a circle centred at the point (a, b) with radius c is:

• (x – a)² + (y – b)² = c²

This can also be written in the form:

x² + y² – 2fx –2gy + h² = 0

If the circle is written in this form, the easiest way to find the centre and radius is to convert it into the above form by completing the square.

Example

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

Write this as x² + 4x + y² – 6y – 12 = 0

Now, (x + 2)² = x² + 4x + 4, so x² + 4x = (x + 2)² – 4, giving us:
(x + 2)² + y² – 6y – 16 = 0
(x + 2)² + (y – 3)² – 25 = 0
(x + 2)² + (y – 3)² = 5²

So the circle has centre (-2, 3) and radius 5