Let S be the set of all a ∈ N such that the area of the triangle formed by the tangent at the point P ( b, c), b, c ∈ N, on the parabola y^2=2 ax and…

Let S be the set of all $\displaystyle \mathrm{a} \in \mathbb{N}$ such that the area of the triangle formed by the tangent at the point $\displaystyle \mathrm{P}(\mathrm{b}$, c), $\displaystyle \mathrm{b}, \mathrm{c} \in \mathbb{N}$, on the parabola $\displaystyle y^2=2 \mathrm{ax}$ and the lines $\displaystyle x=\mathrm{b}, y=0$ is $\displaystyle 16$ unit $\displaystyle ^2$, then $\displaystyle \sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}$ is equal to $\displaystyle \_\_\_\_$.