Tangent equations in various forms:

Point form at (x_1, y_1): T = xx_1/a^2 - yy_1/b^2 - 1 = 0.

Parametric form at (asec(theta), btan(theta)): xsec(theta)/a - ytan(theta)/b = 1.

Slope form: y = mx +/- sqrt(a^2m^2 - b^2). The tangent exists (is real) only when a^2m^2 - b^2 >= 0, i.e., |m| >= b/a. This means no tangent line with slope |m| < b/a touches the hyperbola -- such lines intersect the asymptotes but miss the curve.

Pair of tangents from external point (h, k): T^2 = S * S_1 where S = x^2/a^2 - y^2/b^2 - 1, S_1 = h^2/a^2 - k^2/b^2 - 1, T = xh/a^2 - yk/b^2 - 1.

Chord of contact from (h, k): T = 0, i.e., xh/a^2 - yk/b^2 = 1.

Normal equations:

Point form at (x_1, y_1): a^2x/x_1 + b^2y/y_1 = a^2 + b^2 = c^2.

Parametric form: axcos(theta) + bycot(theta) = a^2 + b^2.

Slope form: y = mx -/+ m(a^2 + b^2)/sqrt(a^2 - b^2*m^2). At most 4 normals can be drawn from an external point.

Key tangent properties: (1) The portion of a tangent between the asymptotes is bisected at the point of tangency. (2) The area of the triangle formed by any tangent and the asymptotes is constant = ab. (3) If the tangent at P meets the asymptotes at Q and R, then PQ = PR.

JEE focus: Tangent condition problems (finding common tangents to a hyperbola and another conic), and the reflection property are the most tested aspects.