The Lesson
The equation of a circle, with a centre with Cartesian coordinates (a, b) is in the form:In this equation,
 x and y are the Cartesian coordinates of points on the (boundary of the) circle.
 a and b are the Cartesian coordinates of the centre of the circle.
 r is the radius of the circle.
Real Examples of Equations of Circles (Not Centred on the Origin)
It is easier to understand the equation of a circle with examples.
A circle centred at (2, 3) with a radius of 5 will have the equation:

A circle centred at (−1, 1) with a radius of 3 will have the equation:
Beware
The Center of a Circle Has Negative Coordinates
The equation of a circle is:The center is (a, b).
 The number being subtracted from the x in the brackets is the xcoordinate of the center.
 The number being subtracted from the y in the brackets is the ycoordinate of the center.
(x − −1)^{2} + ...
Remember, that subtracting a negative number is the same as adding the positive number:
(x − −1)^{2} = (x + 1)^{2}
A negative coordinate will have a + sign in front of it.
A positive coordinate will have a − sign in front of it.
Equations That Don't Quite Look Right
Don't be confused if you see an equation which looks like this:
(x − 1)^{2} + (y − 3)^{2} − 49 = 0
This is still an equation of a circle, as can be seen with a little rearranging:
(x − 1)^{2} + (y − 3)^{2} = 49
Note
Circle Centered at the Origin
A circle centered at the origin has a centre at (0, 0). If it has a radius r, the equation is:(x − 0)^{2} + (y − 0)^{2} = r^{2}
x^{2} + y^{2} = r^{2}