Answer
The center is $(0,1,2)$ and the radius is $\frac{5}{\sqrt{3}}$.
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.
$3x^2+3y^2+3z^2=10+6y+12z$
$(x^2)+(y^2-2y)+(z^2-4z)=\frac{10}{3}$ (rearrange terms)
$(x^2)+(y^2-2y+1)+(z^2-4z+4)=\frac{10}{3}+1+4$ (complete the square)
$(x)^2+(y-1)^2+(z-2)^2=\frac{25}{3}$
Now that the equation is in the form:
$(x−h)^2+(y−k)^2+(z−l)^2=r^2$
The center is $(0,1,2)$ and the radius is $\frac{5}{\sqrt{3}}$.
