Chapter B, Problem 40E

To determine

To calculate: To show that the equation represents the circle and then find the center and radius of circle

Answer to Problem 40E

Equation is equation of circle. Also center of circle is $\left(0,-3\right)$ and radius of circle is $\sqrt{7}$

Explanation of Solution

Given information: Equation is $x^2+y^2+6y+2=0$

Formula Used:

Equation of circle with center as $\left(h,k\right)$ and radius $r$ is given as

  ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$

Calculation:

Equation is given as follows:

  $x^2+y^2+6y+2=0$

Rearranging the terms of equation,

  $\begin{array}{l}{x}^2+y^2+6y+2=0\\ x^2+\left(y^2+6y+9\right)+2=9\\ x^2+{\left(y+3\right)}^2=9-2\\ x^2+{\left(y+3\right)}^2=7\\ x^2+{\left(y-\left(-3\right)\right)}^2={\left(\sqrt{7}\right)}^2\end{array}$

Comparing the above equation with standard equation of circle ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ ,

  $\begin{array}{l}Centre,\left(h,k\right)=\left(0,-3\right)\\ Radius,r=\sqrt{7}\end{array}$

Conclusion:

Hence, given equation is equation of circle. Also center of circle is $\left(0,-3\right)$ and radius of circle is $\sqrt{7}$