Three tangent forms serve different contexts. Point form (T=0): xx1/$a^2$ + yy1/$b^2$ = 1 when the point of tangency is known. Slope form: y = mx +/- sqrt($a^2$$m^2$ + $b^2$) when the slope is given or the tangent is parallel to a known line; the tangency condition $c^2$ = $a^2$$m^2$ + $b^2$ determines whether a line touches the ellipse. Parametric form: (xcos(theta))/a + (ysin(theta))/b = 1 when the eccentric angle is known. From an external point, exactly two tangents exist (S1 > 0). From a point on the ellipse, exactly one. From an interior point, none. The foot of perpendicular from either focus to any tangent lies on the auxiliary circle. The product of perpendicular distances from both foci to any tangent equals $b^2$. The tangent at (ae, $b^2$/a) (end of latus rectum) simplifies to ex + y = a, connecting eccentricity directly to the tangent equation.