Chord of contact from (x1, y1): T = 0 gives xx1/a^2 - yy1/b^2 = 1. Chord with midpoint (x1, y1): T = S1 gives xx1/a^2 - yy1/b^2 = x1^2/a^2 - y1^2/b^2. Pair of tangents: T^2 = S*S1. These are identical in structure to the ellipse formulas, with the sign of the y^2 term changed throughout. The T=0 framework is universal across all conics.