Parabola Tangent Meeting the Unit Circle

Let $O$ be the origin and let $F$ be the focus of the parabola $C_1:\ x^2=2py$ (with $p>0$). The tangent line to $C_1$ at a point $P$ is also tangent to the circle $C_2:\ x^2+y^2=1$, touching it at the point $Q$.

1. When the line $PQ$ has equation $x-y-\sqrt2=0$, find the equation of the parabola $C_1$.

2. As the positive number $p$ varies, let $S_1$ and $S_2$ be the areas of $\triangle FPQ$ and $\triangle FOQ$ respectively. Find the minimum value of $\dfrac{S_1}{S_2}$.