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^2m^2)x^2 - 2a^2mcx - a^2(c^2 + b^2) = 0.

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

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