Chapter 11.1, Problem 56E

To determine

Tofind:Thecenter and radius of the sphere when the equation of the sphere is $x^2+y^2+z^2-9x=0$ .

Answer to Problem 56E

Thecenter and radius of the sphere is $\left(\frac{9}{2},0,0\right)$ and $\frac{9}{2}$ , respectively.

Explanation of Solution

Given information:

The given equation is $x^2+y^2+z^2-9x=0$ .

Calculation:

Represent the given equation in the standard form of the equation of the sphere.

  $\begin{array}{c}{x}^2+y^2+z^2-9x=0\left\{\text{Add}\frac{81}{4}\text{on}\text{both}\text{sides}\right\}\\ x^2-9x+\frac{81}{4}+y^2+z^2=\frac{81}{4}\\ {\left(x-\frac{9}{2}\right)}^2+y^2+z^2={\left(\frac{9}{2}\right)}^2\end{array}$

Compare the above equation with the standard form of the equation of the sphere where $\left(h,k,j\right)$ is the center and the radius is $r$ .

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

From the above, the center of the sphere is $\left(h,k,j\right)=\left(\frac{9}{2},0,0\right)$ and the radius is $r=\frac{9}{2}$ .

Therefore, thecenter and radius of the sphere is $\left(\frac{9}{2},0,0\right)$ and $\frac{9}{2}$ , respectively.