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.