355. Find the equation of the sphere with center at point $C(a ; b ; c)$ and radius $R$.
Solution. By definition, a sphere is the geometric locus of points equally distant from a given point called the center. Therefore, the distance from any point $M(x ; y ; z)$, lying on the sphere, to the center $C(a ; b ; c)$ is $R$, i.e.,

$$
l^{\prime} \overline{(x-a)^{2}+(y-b)^{2}+(z-c)^{2}}=R
$$

hence

$$
(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=R^{2}
$$

This is the desired equation of the sphere. If the center of the sphere coincides with the origin, then $a=b=c=0$ and the equation of the sphere will be

$$
x^{2}+y^{2}+z^{2}=R^{2}
$$