Question #40550

This question tests the intersection of 3D coordinate geometry and calculus-based range analysis.

The primary trap lies in incorrectly evaluating the range of the quadratic parameter $a$ over the half-open interval $t \in (0, 3]$.

Students often blindly plug in the endpoints $t=0$ and $t=3$ to get $a \in (2, 5)$, missing the vertex at $t=2$ which gives the true closed lower bound $a=1$.

Furthermore, the question demands an understanding of topological closure ($\bar{S}$); the set $\bar{S} \setminus S$ specifically isolates the boundary point generated by the open end of the interval ($a \to 5$).

Selecting the point corresponding to $a=1$ or $a=2$ indicates a failure to distinguish between open and closed boundaries in the locus.