How to show that the locus of the centres of the spheres of a co-axial system is a straight line?

Question

I proceeded like this, Let $(x_1, y_1, z_1)$ be a point on the sphere $S_1$ and $(x_2, y_2, z_2)$ be the point satisfied by the sphere $S_2$ of a coaxial system. Now I need to show that there exists straight line joining these two points. That is I need to prove $x_2-x_1$, $y_2-y_1$, $z_2-z_1$ not all zero. Is this right?. If wrong what how should I proceed.