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

A circle has the equation (x-a) + (y-b)² = r² where a and b are the coordinates of the centre of the circle and r is the radius of the circle. The equation of a circle may also be written in the form x² + y² + dx + ey + f = 0 if you expand the brackets out on the left hand side of the equation.

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

Question 1

A circle has centre (3,8) and a radius of 6. Find the equation of the circle. Write down the equation of the circle in the form x² + y² + 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:

x² + y² + dx + ey + f = 0

(x-3)² + (y-8)² = 6²

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

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

x² -6x +9 + y² -16y + 64 = 36

x² + y² -6x -16y +64 =36

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

x² + y² -6x -16y +28 =0

Question 2

Work out the equation of the circle in the form x² + y² + dx + ey + f = 0 that is shown in the picture.

Answer

First of all write down the centre and radius of the circle:

Centre = (2,-1)

Radius = 3

Next write down the values of a,b and r.

a = 2, b = -1 and r =3.

Now substitute these values into the equation of the circle:

x² + y² + dx + ey + f = 0

(x-2)² + (y--1)² = 3²

(x-2)² + (y + 1)² = 9

Now multiply out the brackets and write it in the form x² + y² + dx + ey + f = 0

(x-2)(x-2) + (y+1)(y+1) = 9

x² -4x +4 + y² +2y + 1 = 9

x² + y² -4x + 2y + 5 =9

Now finally take 9 off both sides to get it into the required form.

x² + y² -4x + 2y – 4 =0

Extra help:

Easier circle questions.

Working out the radius and centre of the circle from the equation.

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