Question

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

$x^2+y^2+12x-14y+21=0$

Short answer

(a) Center of the circle$x^2+y^2+12x-14y+21=0$islocalid="1645125277424" $(-6,7)$and radius is $8$

(b) Graph of the circle is

Step-by-step solution

Step 1. given data 

The given equation of the circle is

$x^2+y^2+12x-14y+21=0$

Step 2. Part (a) 

Group the x-terms and y-terms

$$\begin{gathered} x^2+y^2+12x-14y+21=0 \\ \left(x^2+12x\right)+\left(y^2-14y\right)=-21 \end{gathered}$$

complete the squares

$$\begin{gathered} \left(x^2+12x+36\right)+\left(y^2-14y+49\right)=-21+36+49 \\ {\left(x+6\right)}^2+{\left(y-7\right)}^2=64 \end{gathered}$$

Compare the equation with the standard equation of the circle

$$\begin{gathered} {\left(x-(-6)\right)}^2+{\left(y-7\right)}^2=8^2 \\ {\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2 \end{gathered}$$

so center$(h,k)=(-6,7)$

radius$r=8$

Step 3. Part (b) 

Plot the circle of center $(-6,7)$and radius$8$