Graphs

YOU ARE LEARNING:

Equation of a Circle 1

Circles can be plotted on a graph from an equation in the same way as other functions. Finding the equation involves competing the square.

Mathematicians have studied circles for centuries and you will have met them before. It is possible to have an equation for a circle if we know its radius and the coordinate of its centre.

A circle has the following formula, where r is the radius and $(a, b)$ is its centre. $(x - a)^2 + (y - b)^2 = r^2$

The following circle has a radius of 10. What is the coordinate of the centre? $(x+5)^2 + (y -3)^2 = 100$

The following is a circle with centre (1,3). What is the radius? $(x -1)^2 + (y - 3)^2 = 4$

Being able to complete the square will be a key skill to complete this section. It helps us to convert an equation in another form into the general form for a circle.

If you are coming across this for the first time, check out our lesson on completing the square, it'll help you understand the next bit.

For example, let's try completing the square to convert the following equation to the general form for a circle. $x^2 + y^2 + 2x + 4y = 20$

Arrange so that $x$ and $y$ terms are together.

Let's start with $x$

We need to halve the coefficient of $x$ , in this case 2, and place inside brackets with $x$ . $(x + 1)^2$

Multiply out the brackets

$(x+1)(x+1)=x^2+2x+1$ . This is $1$ larger than our original

So, we need to take $1$ away

$(x+1)^2-1$

Complete the square for $y^2+4y$

Nice! Now putting them together...

Our equation becomes: $(x + 1)^2 -1 +(y+2)^2 - 4 = 20$

Collect the numbers together for $(x + 1)^2 -1 +(y+2)^2 - 4 = 20$ .

There we go!

Here is our equation in the general form, with radius 5. $(x+1)^2+(y+2)^2=25$

Find the centre: $(x+1)^2+(y+2)^2=25$

Complete the square for the terms in the square bracket and then give the centre and radius of the circle: $(x^2 + 2x -8) + (y-7)^2 = 0$