Introduction and Key Concepts

This problem asks us to find a condition on the coefficients of a linear equation, given that the line it represents passes through the intersection points of two parabolas. To solve this, we will combine several fundamental concepts from coordinate geometry and algebra:

1. Intersection of Curves: To find where two curves meet, we solve their equations simultaneously. The resulting $(x, y)$ pairs are the coordinates of their intersection points.
2. Condition for a Point on a Line: If a line passes through a specific point, the coordinates of that point must satisfy the equation of the line. Substituting these coordinates into the line's equation will lead to a true statement or a condition on the line's parameters.
3. **Fundamental Property of Real