I was studying about spheres in analytic geometry. I encountered a topic focussing upon radical line. Now, from my knowlege of Euclidean geometry, I know the definition a power of a point with respect to a circle i.e
The power of a point $P$ with respect to a given circle, is the relative distance from the point $P$ to the point in the circle, where, the tangent passing through $P$ intersects the circle, say at $Q$. Then, power of the point $P$ with respect to this circle is $PQ$.
With the aid of this definition, I know the definition of a radical line(also called radical axis in Euclidean Geometry if I am not mistaken) with respect to two non-concentric circles as follows:
The radical axis of two non-concentric circles is the set of points whose power with respect to the circles are equal.
However, I dont have any idea what is meant by a radical axis in case of spheres. I found a definition from the internet, but it mysteriously focussed upon the definition of a radical axis with respect to $\color{green}{\text{three}}$ spheres. And I dont have any idea why are they defining radical axis with the help of three spheres. Is it impossible somehow, to define it for a single or two spheres? Apart from it, tge definition I found goes as follows:
The three radical planes of three spheres intersect in a line. This line is called the radical line of three spheres.
First of all, how do they know that three radical planes of three spheres intersect in a line?
Next, they went on with some calculations of the equation of the so-mentioned line, which was written as follows:
If $S_1=0,S_2=0,S_3=0$ be the equation of three spheres respectively. The equations of their radical planes are $$\begin{align}S_1-S_2=0,S_2-S_3=0,S_1-S_3=0\end{align}$$ and they meet in the line $S_1=S_2=S_3$ iff $S_1-S_2=0,S_2-S_3=0.$ This line is called the radical line of three spheres.
Here, I dont get how they conclude :they meet in the line $S_1=S_2=S_3$ iff $S_1-S_2=0,S_2-S_3=0$?
I am not quite getting it.
Let the 3 spheres $S_k$ have centers $C_k(a_k,b_k,c_k)$ and radii $r_k$ ($k=1,2,3$) with equations
$$(S_k) : \ \ \ x^2+y^2+z^2-2a_kx-2b_ky-2c_kz+d_k=0$$
We agree on the fact that the equation of radical plane of 2 spheres, say $S_1$ and $S_2$ is obtained by taking the difference of the two equations, leaving only first degree terms : this is what is meant by $S_2-S_1=0$ (I prefer to write the the final $= 0$).
The point is that, when you consider the equations of the 3 radical planes :
$$\begin{cases}R_{1,2} \ & \ S_2-S_1&=&0\\ R_{2,3} \ & \ S_3-S_2&=&0\\ R_{3,1} \ & \ S_1-S_3&=&0\end{cases} \ \iff \ \begin{cases}R_{1,2} \ & \ S_2-S_1&=&0\\ R_{2,3} \ & \ S_3-S_2&=&0\end{cases} \ \iff \ S_1=S_2=S_3$$
Indeed, and this answers your question : the third equation is redundant ; it can be retrieved by adding the two first ones. Therefore it can be suppressed. The two remaining equations account for the intersection of 2 planes, i.e., a straight line which is the looked-for radical axis.
A geometric view can help :
The radical plane of spheres $S_p$ and $S_q$ is orthogonal to line segment $[C_pC_q]$ joining their two centers.
These 3 radical planes are all perpendicular to the plane $P$ defined by $C_1,C_2,C_3$, these three planes having a common line... itself orthogonal to plane $P$.