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.