Question

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

$x^2+y^2+6y+5=0$

Short answer

(a) The center and radius of the graph $x^2+y^2+6y+5=0$is $(0,-3)$and $r=2$

(b) The graph of the function is:

Step-by-step solution

Part (a) Step 1. Given

The function $x^2+y^2+6y+5=0$

To find the radius and center of the circle.

Part (a) Step 2. Group x,y-terms

Group the $x-$terms and $y-$terms. Collect the constants on the right side.

$$\begin{gathered} x^2+y^2+6y+5=0 \\ {x}^2+(y^2+6y)=5 \end{gathered}$$

Part (a) Step 3. Complete the squares.

Complete the squares and write in binomial form.

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

Part (a) Step 4. Identify the center and radius.

Identify the center and radius by comparing the equation with standard form of circle ${(x-h)}^2+{(y-k)}^2=r^2$.

From the equation, the center is $(0,-3)$and the radius is $2$

Part (b) Step 1. Given

The function $x^2+y^2+6y+5=0$

To graph the function

Part (b) Step 2. Graph the function

Graph the circle with center $(0,-3)$and radius $2$