Describing a Subset of a Hilbert Space $H$

Question

Let $H$ be a Hilbert space. How can we describe the set $\{ x \in H \mid \|x-y\| = a \|x-z\| \},$ where $y, z \in H$ are fixed and $a > 0$? Geometrically how does it look like?

Answer 1

Hint:

$\|x-y\|$ denotes that distance from the point $x$ to the point $y$.

$\|x-z\|$ denotes that distance from the point $x$ to the point $z$.

Answer 2

It look like the hyperplane parpendicular to the line joining $y$ and $z$ and dividing the line in the proportion $a:(1-a)$.

Answer 3

Square the inner products and work things out. Then, for $a≠1$ and modulo computational mistakes: \begin{equation*} \parallel x-\frac{y-a^{2}z}{1-a^{2}}\parallel =c, \end{equation*} where $c$ is a constant.