This problem is a classic example of how the parametric form of a line can be a highly efficient and powerful tool for solving problems involving distances from a fixed point to the intersection points of a line and a conic section. This method allows us to directly obtain the distances as roots of a quadratic equation, simplifying calculations significantly, especially when dealing with products or sums of these distances.

1. Key Concept: Parametric Representation of a Line and its Intersection with a Conic

Consider a line passing through a fixed point $P(x_1, y_1)$. If this line makes an angle $\theta$ with the positive x-axis, any point $(x, y)$ on this line can be represented in its parametric form as: $\frac{x - x_1}{\cos \theta} = \frac{y - y_1}{\sin \theta} = r$ From this, we can express $x$ and $y$ coordinates in terms of $r$: $x = x_1 + r \cos \theta$ $y = y_1 + r \sin \theta$ Here, $r$ represents the directed distance from the fixed point $P(