Question

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 \(c=1 ?\)