# Circle equations. Using the centre and radius to write down the equation of a circle.

A circle has the equation (x-a) + (y-b)? = r? where a and bare the coordinates of the centre of the circle and r is the radius of thecircle. The equation of a circle may also be written in the form x? + y? + dx +ey + f = 0 if you expand the brackets out on the left hand side of theequation.

Let’s take a look at a few questions on working out theequation of a circle:

**Question 1**

A circle has centre (3,8) and a radius of 6. Find theequation of the circle. Write down the equation of the circle in the form x? +y? + dx + ey + f = 0.

Answer

First of all write down the values of a,b and r.

a = 3, b = 8 and r =6.

Now substitute these values into the equation of the circle:

x? + y? + dx + ey + f = 0

(x-3)? + (y-8)? = 6?

All you need to do is expand all the bracket and simplifyyour answer.

(x-3)(x-3) + (y-8)(y-8) = 36

x? -6x +9 + y? -16y + 64 = 36

x? + y? -6x -16y +64 =36

Now finally take 36 off both sides to get it into therequired form.

x? + y? -6x -16y +28 =0

**Question 2**

Work out the equation of the circle in the form x? + y? + dx+ ey + f = 0 that is shown in the picture.