Problem 31

Given the problem, we have $p=2$, so the equation of the directrix of the parabola is $x=-1$.

Since the line through the focus of the parabola $y^{2}=4x$ intersects the parabola at points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$,

it follows that $|AB|=x_{1}+x_{2}+2$.

Given $x_{1}+x_{2}=9$,

thus, $|AB|=x_{1}+x_{2}+2=11$.

Therefore, the correct choice is $\boxed{A}$.

The line through the focus of the parabola $y^{2}=4x$ intersects the parabola at points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$, hence $|AB|=x_{1}+x_{2}+2$, from which the chord length can be easily determined.

This problem tests the simple properties of a parabola. The key to solving the problem is understanding that the distance to the focus is equal to the distance to the directrix. This relationship transforms the problem of finding the chord length into a problem of finding the distance from a point to a line, significantly reducing the difficulty of solving the problem.