Aus Online Mathematik Brückenkurs 1

As the equation stands, it is difficult directly to know anything about the circle, but if we complete the square and combine x- and y-terms together in their own respective square terms, then we will have the equation in the standard form

\displaystyle (x-a)^2 + (y-b)^2 = r^2\,,

and we will then be able to read off the circle's centre and radius.

If we take the x- and y-terms on the left-hand side and complete the square, we get

\displaystyle \begin{align} x^2 + 2x &= (x+1)^2-1^2\,,\\[5pt] y^2 - 2y &= (y-1)^2-1^2\,, \end{align}

and then the whole equation can be written as

\displaystyle (x+1)^2 - 1^2 + (y-1)^2 - 1^2 = 1\,,

or, with the constants moved to the right-hand side,

\displaystyle (x+1)^2 + (y-1)^2 = 3\,\textrm{.}

This is a circle having its centre at (-1,1) and radius \displaystyle \sqrt{3}\,.