The tangent at (x1,y1) on x^2+y^2=a^2 is xx1+yy1=a^2. For the general circle, tangent at (x1,y1) is xx1+yy1+g(x+x1)+f(y+y1)+c=0. This is the T=0 formula obtained by the substitution rule: replace x^2 with xx1, y^2 with yy1, 2x with (x+x1), 2y with (y+y1). This same substitution gives the chord of contact, chord with midpoint, and pair of tangents.