This problem asks us to determine the locus of the point of intersection of two straight lines whose equations depend on a parameter $t$. The general approach to such problems involves solving the system of equations for the coordinates $(x, y)$ in terms of $t$, and then eliminating $t$ to obtain a direct relationship between $x$ and $y$. This resulting equation will represent the locus.

1. Key Concept: Finding the Locus of Intersection of Parametric Lines

The locus of a point is the path traced by that point as it satisfies certain conditions. When the equations of two lines are given in terms of a parameter, say $t$, their point of intersection $(x, y)$ will have coordinates that are functions of $t$. To find the locus of this point, we