Key Concepts

1. Locus of a Point: The locus of a point is the set of all points that satisfy a given geometric condition. When we talk about the locus of the intersection of two lines, we are looking for the path traced by this intersection point as some underlying parameter (in this case, $k$) changes.

2. Finding the Locus by Eliminating a Parameter: If the equations of two lines (or any two curves) depend on a common parameter, the coordinates $(x, y)$ of their point of intersection will also depend on this parameter. To find the locus of this intersection point, we need to eliminate the parameter from the system of equations. The resulting equation, expressed solely in terms of $x$ and $y$, will represent the desired