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For real $c>0, c \neq 1$, and distinct points $\vec{p}$ and $\vec{q}$ in $\mathbb{R}^{k}$, consider the points $\vec{x} \in \mathbb{R}^{k}$ that satisfy
$\|\vec{x}-\vec{p}\|=c\|\vec{x}-\vec{q}\| .$
Show that these points lie on a sphere, say $\left\|\vec{x}-\vec{x}_{0}\right\|=r$, so the center is at $\vec{x}_{0}$ and the radius is $r$. Thus, find center and radius of this sphere in terms of $\vec{p}, \vec{q}$ and $c$. What if $c=1 ?$
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