For the hyperbola $x^2$/$a^2$ - $y^2$/$b^2$ = 1 and a point (h, k), compute $S_1$ = $h^2$/$a^2$ - $k^2$/$b^2$ - 1. If $S_1$ > 0, the point is outside the curve (in the region not between the branches). If $S_1$ < 0, the point is inside (between the branches or on the conjugate axis side). If $S_1$ = 0, the point is on the curve.

Caution: "Inside" and "outside" for a hyperbola are counterintuitive compared to the ellipse. The region between the two branches (where $S_1$ < 0) is technically "inside," while the regions beyond the vertices ($S_1$ > 0) are "outside."

For a line y = mx + c and the hyperbola $x^2$/$a^2$ - $y^2$/$b^2$ = 1: substitute to get ($b^2$ - $a^2$*$m^2$)$x^2$ - 2$a^2$*mcx - $a^2$($c^2$ + $b^2$) = 0.

The line intersects the hyperbola in: (1) Two points if $b^2$ - $a^2$$m^2$ != 0 and D > 0. (2) One point (tangent) if D = 0. (3) One point (parallel to asymptote) if $b^2$ - $a^2$$m^2$ = 0, i.e., m = +/- b/a. (4) No real points if D < 0.

The tangent condition $c^2$ = $a^2$$m^2$ - $b^2$ follows from D = 0. Unlike the ellipse where $c^2$ = $a^2$$m^2$ + $b^2$ (always positive), here $c^2$ can be negative in the formula, meaning |m| must be large enough for tangency.