The general equation of a circle whose center is located at the point \(\left( {h,k} \right)\) and whose radius is \(r\) is given by

\({\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}\)

Now we can just plug-n-chug this formula to write the equations of any circle we want.

Example:Write the equation of a circle whose center is \(\left( {6, - 6} \right)\) and with circumference \(2\pi \sqrt {62} \).

Solution:  We have the coordinates of the center of the circle.  Now we simply need to break down the circumference and figure out what the radius is.  We know that

\(Circumference = 2\pi r\)

so that

\(2\pi r = 2\pi \sqrt {62} \)

\(r = \sqrt {62} \)

So the radius is \(\sqrt {62} \). We plug-n-chug:

\({\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}\)

\({\left( {x - 6} \right)^2} + {\left( {y - \left( { - 6} \right)} \right)^2} = {\left( {\sqrt {62} } \right)^2}\)

\({\left( {x - 6} \right)^2} + {\left( {y + 6} \right)^2} = 62\)

Then the general equation of the circle with the information given is

\({\left( {x - 6} \right)^2} + {\left( {y + 6} \right)^2} = 62\)

EXAMPLE:  Write the equation of the circle with center \(\left( {4,6} \right)\) and circumference \(8\pi \).

SOLUTION:  We have

\(circumference = 2\pi r\)

so that

\(2\pi r = 8\pi \)

\(r = 4\)

Then the equation is

\({\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}\)

\({\left( {x - 4} \right)^2} + {\left( {y - 6} \right)^2} = 16\)

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