Aus Online Mathematik Brückenkurs 1

If the circle is to contain the point \displaystyle \left( -1 \right.,\left. 1 \right), then that point's distance away from the centre \displaystyle \left( 2 \right.,\left. -1 \right) must equal the circle's radius, \displaystyle r. Thus, we can obtain the circle's radius by calculating the distance between \displaystyle \left( -1 \right.,\left. 1 \right) and \displaystyle \left( 2 \right.,\left. -1 \right) using the distance formula:

\displaystyle \begin{align} & r=\sqrt{\left( 2-\left( -1 \right) \right)^{2}+\left( -1-1 \right)^{2}}=\sqrt{3^{2}+\left( -2 \right)^{2}} \\ & =\sqrt{9+4}=\sqrt{13} \\ \end{align}

When we know the circle's centre and its radius, we can write the equation of the circle,

\displaystyle \left( x-2 \right)^{2}+\left( y-\left( -1 \right) \right)^{2}=\left( \sqrt{13} \right)^{2}

which the same as

\displaystyle \left( x-2 \right)^{2}+\left( y+1 \right)^{2}=13

NOTE: A circle having its centre at \displaystyle \left( a \right.,\left. b \right) and radius \displaystyle r has the equation

\displaystyle \left( x-a \right)^{2}+\left( y-b \right)2=r^{2}