Finding the vertex of a cone

Question

Let a 3D, circular cone of vertex $\vec{v}=[v_x,v_y,v_z]$ given by the equation: $$ (x-v_x)^2+(y-v_y)^2=(z-v_z)^2 $$ The equation of this cone can also be written as: $$ (\vec{x}-\vec{v})^TA(\vec{x}-\vec{v})=0, \textrm{ with } A=\begin{bmatrix}1&0&0\\0&1&0\\0&0&-1\end{bmatrix} $$ Given a set of measurements $\begin{Bmatrix}{\vec{x_1},\dotsc,\vec{x_n}}\end{Bmatrix}$ of points that should belong to this cone, how to find analytically the vertex $\vec{v}$?