Strategy 1 - Identify the form first: Determine if the hyperbola is standard, conjugate, rectangular, or shifted. The rectangular hyperbola xy = $c^2$ has its own set of parametric formulas that are much simpler. Converting to the appropriate form saves significant computation.

Strategy 2 - Use parametric equations for curve-point problems: When dealing with tangent, normal, chord, or locus problems, parametric representation (asec(theta), btan(theta)) or (ct, c/t) for rectangular hyperbola makes algebra manageable. The parameter also helps in applying cyclic point conditions.

Strategy 3 - Asymptote relationships: The equation of a conic sharing asymptotes with $x^2$/$a^2$ - $y^2$/$b^2$ = 1 is $x^2$/$a^2$ - $y^2$/$b^2$ = k. Use this to find the conjugate hyperbola (k = -1) or to construct a hyperbola passing through a given point.

Strategy 4 - Comparison with ellipse: Most hyperbola formulas mirror ellipse formulas with sign changes. Key differences: $c^2$ = $a^2$ + $b^2$ (not $a^2$ - $b^2$), tangent condition has minus ($a^2$$m^2$ - $b^2$ not $a^2$$m^2$ + $b^2$), and director circle has minus ($a^2$ - $b^2$ not $a^2$ + $b^2$).

Strategy 5 - Common tangent problems: Finding common tangents to two conics (e.g., hyperbola and ellipse, or hyperbola and parabola) requires equating the tangent condition from both conics to find common values of m and c.

Strategy 6 - Eccentricity relationships: For conjugate hyperbolas, 1/$e^2$ + 1/e'^2 = 1. For confocal conics (same foci), the tangent at their intersection point is perpendicular.

JEE typically tests: tangent/normal equations, asymptote properties, eccentricity problems, and rectangular hyperbola problems involving concyclic points.