  Given a set $S$ of $n$ static points and a free point $p$ in the Euclidean
plane, we study a new variation of the minimum enclosing circle problem, in
which a dynamic weight that equals to the reciprocal of the distance from the
free point $p$ to the undetermined circle center is included. In this work, we
prove the optimal solution of the new problem is unique and lies on the
boundary of the farthest-point Voronoi diagram of $S$, once $p$ does not
coincide with any vertex of the convex hull of $S$. We propose a tree structure
constructed from the boundary of the farthest-point Voronoi diagram and use the
hierarchical relationship between edges to locate the optimal solution. The
plane could be divide into at most $3n-4$ non-overlapping regions. When $p$
lies in one of the regions, the optimal solution locates at one node or lies on
the interior of one edge in the boundary of the farthest-point Voronoi diagram.
Moreover, we apply the new variation to calculate the maximum displacement of
one point $p$ under the condition that the displacements of points in $S$ are
restricted in 2D rigid motion.
