Chord of contact: The chord of contact of tangents from an external point (h, k) to $x^2$/$a^2$ - $y^2$/$b^2$ = 1 is T = 0: xh/$a^2$ - yk/$b^2$ = 1.

Polar of a point: The polar of point (h, k) with respect to the hyperbola is also T = 0: xh/$a^2$ - yk/$b^2$ = 1. If (h, k) is outside the curve, the polar is the chord of contact. If (h, k) is on the curve, the polar is the tangent at that point. If (h, k) is inside, the polar doesn't touch the curve but has geometric significance.

Pole-polar duality: If the polar of P passes through Q, then the polar of Q passes through P (conjugate points). This reciprocal relationship is a projective property.

Chord with given midpoint (h, k): The equation is T = $S_1$, i.e., xh/$a^2$ - yk/$b^2$ - 1 = $h^2$/$a^2$ - $k^2$/$b^2$ - 1, simplifying to xh/$a^2$ - yk/$b^2$ = $h^2$/$a^2$ - $k^2$/$b^2$.

Director circle (locus of perpendicular tangent intersection): $x^2$ + $y^2$ = $a^2$ - $b^2$. Exists only when a > b. For a = b (rectangular hyperbola), perpendicular tangents intersect at the center. For a < b, no pair of perpendicular tangents exists.

These concepts parallel the ellipse but with crucial sign differences (minus instead of plus in the conic equation). JEE problems often test whether students correctly handle these sign changes.