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

Given that $P\left(b,c\right)$ lies on the parabola. So $c^2=2ab$

Now equation of tangent ot the parabola $y^2=2ax$ in point form $y{y}_1=a\left(x+x_1\right)$

it implies that $cy=a\left(x+b\right)$

for the point $B$, put $y=0$, hence $x=-b$

So , the area of triangle PBA is $16$ square units

it implies that $bc=16$

Since the values are all natural numbers, and $c^2=2ab$, the possible values of $a$ are 128,16 and 2