real analysis - Verify if $n$ dimensional spheres intersect

Consider $x_1,\dots,x_n$ vectors lying in $\mathbb{R}^n$. Also, consider positive numbers $\alpha_1,\dots,\alpha_n$. Consider $n$ spheres with center as $x_i$ and radii $\alpha_i$. How do I check the set of solutions to the set of equations
$$\begin{align} ||x-x_i||_2 = \alpha_i~,~ \forall i \end{align}$$
is non-empty. Is this problem studied in literature?

Answer

The question is well-studied. For example, you can read this.

One of the methods works like this:
Rewrite your equation as
$$2x_ix = x^2 +x_i^2 - \alpha_i^2.$$
Find $x$ in form $x=ru+v$, where $r=x^2$, $u$ is the solution of $2x_iu=1$, $v$ is the solution of $2x_iv=x_i^2-\alpha_i^2$. Then
$$r=x^2=(ru+v)^2=r^2u^2+2uv r+v^2$$
is a quadratic equation on $r$. If this equation has roots, then your $n$ spheres have a common point.