Now that we know what the equation of a circle means, we can use it to identify the center and the radius and sketch the graph of the circle in the plane.  But just for a refresher, let’s restate the definition of the equation of a circle.

DEFINITION:  The equation of a circle with center \(\left( {h,k} \right)\) and radius \(r\) is given by

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

Let’s implement this information now.

EXAMPLE:  Identify the center and radius of the circle.  Then sketch the graph of the circle.

\({\left( {x - \frac{1}{2}} \right)^2} + {\left( {y + \frac{1}{2}} \right)^2} = 4\)

SOLUTION:  We manipulate this equation slightly to get it exactly in the form above:

\({\left( {x - \frac{1}{2}} \right)^2} + {\left( {y + \frac{1}{2}} \right)^2} = 4\)

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

Now this equation is in the general form.  From here we can conclude that

\({\text{Radius}} = 2\)

\({\text{Centeris}}\left( {\Large \frac{1}{2}, - \Large \frac{1}{2}} \right)\)

Here is the graph:

EXAMPLE:  Identify the center and radius of the circle.  Then sketch the graph of the circle.

\({\left( {x - 2} \right)^2} + {\left( {y - \frac{7}{2}} \right)^2} = 7\)

SOLUTION:  Again we manipulate the equation into the general form.  We have

\({\left( {x - 2} \right)^2} + {\left( {y - \frac{7}{2}} \right)^2} = 7\)

\({\left( {x - 2} \right)^2} + {\left( {y - \frac{7}{2}} \right)^2} = {\left( {\sqrt 7 } \right)^2}\)

Then the radius of the circle is \(\sqrt 7  \approx 2.6\), and the center is \(\left( {2,\frac{7}{2}} \right)\). The graph is below.

Below you can download some free math worksheets and practice.