Question

In the following exercises, ⓐ identify the center and radius, and ⓑ graph.

$x^2+y^2-6x-8y=0$

Short answer

(a) Center of the circle$x^2+y^2-6x-8y=0$is$(3,4)$and radius is$5$

(b) graph of the circle is

Step-by-step solution

Step 1. given data 

The given equation of the circle is

$x^2+y^2-6x-8y=0$

Step 2. Part (a) 

Group the x-terms and y-terms

$$\begin{gathered} x^2+y^2-6x-8y=0 \\ \left(x^2-6x\right)+\left(y^2-8y\right)=0 \end{gathered}$$

complete the squares

$$\begin{gathered} \left(x^2-6x+9\right)+\left(y^2-8y+16\right)=9+16 \\ {\left(x-3\right)}^2+{\left(y-4\right)}^2=5^2 \end{gathered}$$

Compare the equation with the standard equation of the circle

$$\begin{gathered} {\left(x-3\right)}^2+{\left(y-4\right)}^2=5^2 \\ {\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2 \end{gathered}$$

so center$(h,k)=(3,4)$

radius$r=5$

Step 3. Part (b) 

Plot the circle of center $(3,4)$and radius$5$