## Equation of a Circle

### Equation of a Circle Notes

We find the **equation of a circle** from the coordinates of its centre and its radius. Consider this circle whose **centre is at the point (a,b) **and whose **radius is r**. The equation of the circle is given by

For a particular circle, the values of , and are specified whereas and are left as general points. This is much like when we specify and in when we have the equation of a given straight line. Note that the equation of a straight line in this form is *explicit – *this means that y is the subject and is given in terms of x. The circle equation, on the other hand, is *implicit* – put simply the x and y appear anywhere in the equation.

The derivation of the equation of a circle is from an application of Pythagoras and can be seen by drawing a right-angled triangle such that the radius is the hypoteneuse – see the derivation here. If we pick any point on the circle , we see that the shorter sides have lengths and while the hypoteneuse has length . Since this is true for any point on the circle it follows that .

You will be presented with questions that expect you to know the equation of a circle. Other questions might bring in knowledge from other areas of maths such as finding mid-points. In more complicated questions they may ask you to find gradients using the knowledge that a tangent to a circle is perpendicular to its radius. Before we try some examples, we will look at some of the other things you need to know about circles.

### Things to Learn for Circle Equation Questions

**You should know the following facts:**

- Most importantly, the equation of a circle.
- Secondly, that the angle subtended from a diameter at the circumference is a right angle.
- Next, the radius and tangent touch at right angles to one another.
- Finally, a perpendicular from a chord bisects the chord.

**You should be able to find the equation of a circle:**

- using completing the square and/or Pythagoras,
- given a triangle that has all three points on its circumference and
- find the equation of a tangent using perpendicular gradients.

## Examples

## Videos

Past exam question and solutions on the equation of a circle including the equation of a tangent.

Past AS maths exam question on circle geometry which is also heavy on trigonometry.