<s>[INST] Prob. 19, Chap. 1, in Baby Rudin: For what $\mathbf{c}$ and $r > 0$ does this equivalence hold? Here's Prob. 19 in Chap. 1 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: 

Suppose $\mathbf{a} \in \mathbb{R}^k$, $\mathbf{b} \in \mathbb{R}^k$. Find $\mathbf{c} \in \mathbb{R}^k$ and $r > 0$ such that, for all $\mathbf{x} \in \mathbb{R}^k$, we have  $$\vert \mathbf{x} - \mathbf{a} \vert = 2 \vert \mathbf{x} - \mathbf{b} \vert$$ if and only if $$\vert \mathbf{x} - \mathbf{c} \vert = r.$$

Although Rudin has given a solution, namely $3\mathbf{c} = 4 \mathbf{b} - \mathbf{a}$, $3r =  2\vert \mathbf{b} - \mathbf{a} \vert$, I'm wondering  how he's obtained it. 
How to attack this type of a problem?  
Is this problem part of the exercises by some well-thought-out design? I mean is it going to be used later on in the book? Or, is it just to give the reader some practice? 
 [/INST] First a comment. What this problem really says is that the subset of $\mathbb{R}^k$ defined by the equation $|x - a| = 2|x - b|$ is a sphere. So there is really not going to be any choice in determining $c$ and $r$. The point $c$ will need to be the centre of the sphere and $r$ its radius.
I imagine the main point of this exercise is to give readers practice with dot products, particularly the relation $|x|^2 = x \cdot x$.
To prove the result, write the following succession of equivalent equations, calculations that amount to completing the square with dot products.
$$
\begin{align*}
 2|x - b| &= |x - a| \\
4|x-b|^2 &= |x - a|^2 \\
 4 (x - b) \cdot (x - b) &= (x - a) \cdot (x - a) \\
 4 x \cdot x - 8 x \cdot b + 4 b \cdot b &= x \cdot x - 2 x \cdot a + a \cdot a \\
3 x \cdot x - 8 x \cdot b + 2 x \cdot a &= a \cdot a - 4 b \cdot b \\
x \cdot x - 2 x \cdot \left( \frac{4}{3}b - \frac{1}{3} a\right) &= \frac{1}{3}a \cdot a - \frac{4}{3}b \cdot b \\
x \cdot x - 2 x \cdot \left( \frac{4}{3}b - \frac{1}{3} a\right) +\left( \frac{4}{3}b - \frac{1}{3} a\right) \cdot \left( \frac{4}{3}b - \frac{1}{3} a\right) &= \frac{1}{3}a \cdot a - \frac{4}{3}b \cdot b + \left( \frac{4}{3}b - \frac{1}{3} a\right) \cdot \left( \frac{4}{3}b - \frac{1}{3} a\right)\\
(x - c) \cdot (x -c) &= r^2 \\
|x - c| &= r
\end{align*}
$$
where $c = \frac{4}{3}b - \frac{1}{3} a$ and $r^2 =  \frac{4}{9} a \cdot a - \frac{8}{9} a \cdot b + \frac{4}{9} b \cdot b = \frac{4}{9}(a - b) \cdot (a - b)$, so $r = \frac{2}{3}|a - b|$.
Technically, the exercise is incorrect when $a = b$. 
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