Point form: xx1/$a^2$ - yy1/$b^2$ = 1. Slope form: y = mx +/- sqrt($a^2$$m^2$ - $b^2$), with tangent condition $c^2$ = $a^2$$m^2$ - $b^2$. This requires $a^2$$m^2$ > $b^2$, i.e., |m| > b/a. Slopes with |m| < b/a cannot produce tangents (the line would intersect both branches). Parametric: (xsec(theta))/a - (y*tan(theta))/b = 1. Two tangents can be drawn from any external point. The asymptotes have slope +/-b/a, which is exactly the boundary where no tangent exists.