Trevor Pythagoras

Since a circle is made up of all the points a fixed distance (its radius) from a given point (its centre) then the equation of a circle simply needs to ensure this is true. This can be done using Pythagoras’s theorem. This is because we can draw a right angled triangle with the centre of the circle at one corner and the point on the circle at the opposite corner as shown below. The radius is then the hypotenuse, the vertical side is the difference between the y co-ordinate of the point and that of the centre and the horizontal side is the difference between the x co-ordinate of the point and centre. From Pythagoras we therefore know that a circle of radius r and centre (a,b) must have a Cartesian equation

r² = (x-a)² + (y-b)²

Circle on Cartesian axis

However, we can expand these brackets out to get

r² = x² – 2ax + a² + y² – 2by + b²

but since a²+b²+r², -a and -b are all constant we can let

c = a²+b²-r²,
f = -a
g = -b

to get

x² + y² + 2gx + 2fy + c = 0
where the circle has a centre (-g,-f) and radius √(a²+b²-c²)