Question

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

$x^2+y^2+z^2+4x-6y+2z=10$.

Short answer

The given equation $x^2+y^2+z^2+4x-6y+2z=10$ is the equation of sphere with center $(-2,3,-1)$ and radius $2\sqrt{6}$

Step-by-step solution

Step 1. Given information.

Given: $x^2+y^2+z^2+4x-6y+2z=10$

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

$\begin{array}{c}{x}^2+y^2+z^2+4x-6y+2z=10\\ \left(x^2+4x+4\right)+\left(y^2-6y+9\right)+\left(z^2+2z+1\right)=10+4+9+1\\ {(x+2)}^2+{(y-3)}^2+{(z+1)}^2=24\end{array}$

add samenumbers both side.
Comparing this with standard equation of sphere, we can see that the center is $(-2,3,-1)$and the radius is $2\sqrt{6}$