Question

Center and Radius of a Sphere Show that the equation represents a sphere, and find its center and radius.

$x^2+y^2+z^2-10x+2y+8z=9$.

Short answer

The given equation $x^2+y^2+z^2-10x+2y+8z=9$ is the equation of sphere with center $(5,-1,-4)$ and radius $\sqrt{51}$

Step-by-step solution

Step 1. Given information.

Given: $x^2+y^2+z^2-10x+2y+8z=9$

Step 2. We have to show that the equation represents a sphere, and find its center and radius.

We will complete the squares in x,y and z terms to rewrite the given equation in the form of an equation of sphere.

The given equation is

$\begin{array}{c}{x}^2+y^2+z^2-10x+2y+8z=9\\ \left(x^2-10x+25\right)+\left(y^2+2y+1\right)+\left(z^2+8z+16\right)=9+25+1+16\\ {(x-5)}^2+{(y+1)}^2+{(z+4)}^2=51\end{array}$

add samenumbers both side.
Comparing this with standard equation of sphere, we can see that the center is $(5,-1,-4)$and the radius is $\sqrt{51}$