Answer
The center is (2,0,−6) and the radius is $\sqrt {40\frac{1}{2}}.$.
Work Step by Step
The equation for a sphere is represented by:
$(x−h)^2+(y−k)^2+(z−l)^2=r^2$
in which the point (h,k,l) is the center of the sphere and r is the radius.
To get the given equation into this form, we must complete the square for all three variables.
In order to do this add $(\frac{b}{2})^2$ to both sides of the equation (b is the coefficient before the x, y, or z term). Since there are three different variables, we must complete the square three times.
$2x^2+2y^2+2z^2=8x-24z+1$
$(x^2−4x)+y^2+(z^2+12z)=\frac{1}{2}$ (rearrange terms)
$(x^2−4x+4)+y^2+(z^2+12z+36)=\frac{1}{2}+4+36$
$(x−2)^2+y^2+(z+6)^2=40\frac{1}{2}$
Now that the equation is in the form:
$(x−h)^2+(y−k)^2+(z−l)^2=r^2$
The center is (2,0,−6) and the radius is $\sqrt {40\frac{1}{2}}.$