This problem asks us to find the equation of a chord of a parabola given its midpoint, and then identify which of the provided points lies on this chord. The most efficient and standard method to solve this type of problem is to use the $T=S_1$ formula for conic sections.

1. Key Concept: Equation of a Chord with a Given Midpoint ($T=S_1$)

For any conic section represented by the general equation $S \equiv Ax^2 + By^2 + Cxy + Dx + Ey + F = 0$, the equation of a chord $PQ$ whose midpoint is $(x_1, y_1)$ is given by the formula $T = S_1$. This formula is a powerful shortcut and is fundamental in coordinate geometry for