Answer
$\text{ graph of a circle with center }$ $
(0,4)
\text{ and radius of }
5$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate whether the given equation, $
x^2+y^2-8y-9=0
,$ has a circle as its graph, change the form of the equation to the Center-Radius Form. Then evaluate the value of $r^2.$ If the value of $r^2$ is positive, then the graph of the equation is a circle. If $r^2$ is equal to zero, then the equation describes a point. If $r^2$ is negative, the graph is nonexistent.
$\bf{\text{Solution Details:}}$
Grouping the $x$-variable and the $y$-variables and transposing the constant, the given equation is equivalent to
\begin{array}{l}\require{cancel}
x^2+(y^2-8y)=9
.\end{array}
Since the coefficient of $y$ is $
-8
,$ add $\left(\dfrac{
-8
}{2}\right)^2=
16
$ to both sides of the equation to complete the square for $x$. That is,
\begin{array}{l}\require{cancel}
x^2+(y^2-8y+16)=9+16
\\\\
x^2+(y-4)^2=25
.\end{array}
In the form $(x-h)^2+(y-k)^2=r^2,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
(x-0)^2+(y-(4))^2=5^2
.\end{array}
Since $r=5,$ is greater than zero, then the equation has a $\text{ graph of a circle with center }$ $
(0,4)
\text{ and radius of }
5
.$
