A circle centrein theplane and radiushas a cartesian equation given byThis can be seen from the graph below.

We obtain the equationfor the circle by applying Pythagoras theorem to the right – angled triangle shown.

In general with can define a a circle by three points which lie on it, a radius and two points which lie on it, if we have the endpoints of the diameter or the radius... A wide range of questions may be asked.

Example. Find the equation of the circle with the diameter whose endpoints are (3,1) and (9,5).

The centre of the circle is and the midpoint of the diameter =

The length of the diameter is the distance between (3,1) and (9,5), and having found this distance, we can halve it to get the radius.

The equation is