Chord of contact from (x1,y1): T = 0 => xx1/a^2 + yy1/b^2 = 1. Chord with midpoint (x1,y1): T = S1 => xx1/a^2 + yy1/b^2 - 1 = x1^2/a^2 + y1^2/b^2 - 1, simplifying to xx1/a^2 + yy1/b^2 = x1^2/a^2 + y1^2/b^2. Pair of tangents: T^2 = S*S1.