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.