4.17. A sphere passes through the point $A(4,2,2)$ and has its center at the point $C(1,-1,-1)$. Write its equation.
Solution. The equation of a sphere centered at the origin has the form

$$
\left(x^{\prime}\right)^{2}+\left(y^{\prime}\right)^{2}+\left(z^{\prime}\right)^{2}=R^{2}
$$

Since the center of the given sphere is located at the point with coordinates $C(1,-1,-1)$, it is natural to perform a parallel translation of the coordinate system so that the new origin is at point $C$. Then the old coordinates $x$, $y, z$ will be related to the new ones $x^{\prime}, y^{\prime}, z^{\prime}$ as follows:

$$
x^{\prime}=x-1, y^{\prime}=y+1, \quad z^{\prime}=z+1
$$

Therefore, in the old coordinates, the equation of the sphere is

$$
(x-1)^{2}+(y+1)^{2}+(z+1)^{2}=R^{2}
$$

To find the radius, substitute the coordinates of point $A(4,2,2)$ into the equation of the sphere:

$$
(4-1)^{2}+(2+1)^{2}+(2+1)^{2}=R^{2}, \quad R^{2}=27, \quad R=\sqrt{27}
$$