Chapter B, Problem 39E

To determine

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

Answer to Problem 39E

Equation is equation of circle. Also center of circle is $\left(2,-5\right)$ and radius of circle is $4$

Explanation of Solution

Given information: Equation is $x^2+y^2-4x+10y+13=0$

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:

  $x^2+y^2-4x+10y+13=0$

Rearranging the terms of equation,

  $\begin{array}{l}{x}^2-4x+y^2+10y+13=0\\ \left(x^2-4x+4\right)+\left(y^2+10y+25\right)+13=4+25\\ {\left(x-2\right)}^2+{\left(y+5\right)}^2=29-13\\ {\left(x-2\right)}^2+{\left(y-\left(-5\right)\right)}^2=16\\ {\left(x-2\right)}^2+{\left(y-\left(-5\right)\right)}^2=4^2\end{array}$

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

  $\begin{array}{l}Centre,\left(h,k\right)=\left(2,-5\right)\\ Radius,r=4\end{array}$

Conclusion:

Hence, given equation is equation of circle. Also center of circle is $\left(2,-5\right)$ and radius of circle is $4$