Question

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

$x^2+y^2+2x+6y+9=0$

Short answer

(a) Center of the circle$x^2+y^2+2x+6y+9=0$is $(-1,-3)$and radius is $1$

(b) graph of the circle is

Step-by-step solution

Step 1. given data 

The given equation of the circle is

$x^2+y^2+2x+6y+9=0$

Step 2. Part (a) 

Group the x-terms and y-terms

$$\begin{gathered} x^2+y^2+2x+6y+9=0 \\ (x^2+2x)+(y^2+6y)=-9 \end{gathered}$$

complete the squares

role="math" localid="1645116279245" $$\begin{gathered} (x^2+2x+1)+(y^2+6y+9)=-9+9+1 \\ {(x+1)}^2+{(y+3)}^2=1 \end{gathered}$$

Compare the equation with the standard equation of the circle

$$\begin{gathered} {(x-(-1\left)\right)}^2+{(y-(-3\left)\right)}^2=1^2 \\ {(x-h)}^2+{(y-k)}^2=r^2 \end{gathered}$$

so center$(h,k)=(-1,-3)$

radius$r=1$

Step 3. Part (b) 

Plot the circle of center $(-1,-3)$and radius$1$