Tangent equations in various forms:

Point form at ($x_1$, $y_1$): T = x*$x_1$/$a^2$ - y*$y_1$/$b^2$ - 1 = 0.

Parametric form at (asec(theta), btan(theta)): xsec$\frac{theta}{a}$ - ytan$\frac{theta}{b}$ = 1.

Slope form: y = mx +/- sqrt($a^2$$m^2$ - $b^2$). The tangent exists (is real) only when $a^2$$m^2$ - $b^2$ >= 0, i.e., |m| >= $\frac{b}{a}$. 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 = $\frac{xh}{a}$^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 ($x_1$, $y_1$): $a^2$*x/$x_1$ + $b^2$*y/$y_1$ = $a^2$ + $b^2$ = $c^2$.

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

Slope form: y = mx -/+ m$\frac{a^2 + b^2}{sqrt}$($a^2$ - $b^2$*$m^2$). 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.