Key Concept: Length of a Chord using Distance Formula and Vieta's Formulas

To find the length of a chord connecting two points $(x_1, y_1)$ and $(x_2, y_2)$ on a curve, we use the distance formula: $L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ A highly efficient strategy for problems involving chords of conic sections (like parabolas) is to find the intersection points by solving the curve and line equations simultaneously. Instead of explicitly calculating the coordinates of these intersection points, we can leverage Vieta's formulas to find the sum and product of the roots of the resulting quadratic equation. These values then allow us to compute the squared differences $(x