A parabola $\mathcal P$ has its vertex at the origin $(0,0)$ and its focus at $(0, 2)$. Let $A$ and $B$ be points on $\mathcal P$ such that $\angle AV_1B = 90^\circ$, where $V_1$ is the vertex of $\mathcal P$. The locus of the midpoint of $\overline{AB}$ is another parabola $\mathcal Q$. Determine the ratio $\frac{F_1F_2}{V_1V_2}$, where $F_1$ and $F_2$ are the foci of parabolas $\mathcal P$ and $\mathcal Q$, respectively, and $V_1$ and $V_2$ are their vertices. Let's think step by step and output the final answer within \boxed{}.
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