The rectangular hyperbola xy = c^2 is the most common non-standard form in JEE. It is obtained by rotating x^2 - y^2 = 2c^2 by 45 degrees. Its eccentricity is sqrt(2), and its asymptotes are the coordinate axes.

Parametric form: (ct, c/t) where t != 0. This elegant parametrization simplifies all computations.

Tangent at (ct, c/t): x/(ct) + y/(c/t) = 2, or x/t + yt = 2c. Equivalently: x + yt^2 = 2ct.

Normal at (ct, c/t): xt^3 - yt = c(t^4 - 1), or xt^3 - yt - ct^4 + c = 0.

Chord joining parameters t_1 and t_2: x + yt_1t_2 = c(t_1 + t_2).

Key results for rectangular hyperbola:

• Four concyclic points: If t_1, t_2, t_3, t_4 are parameters of four concyclic points, then t_1t_2t_3*t_4 = 1.
• Orthocentric system: The orthocentre of a triangle inscribed in xy = c^2 with parameters t_1, t_2, t_3 is the point (-c/(t_1t_2t_3), -ct_1t_2t_3). If the fourth point of the concyclic set is t_4 = 1/(t_1t_2*t_3), the orthocentre lies on the curve.
• Normal at parameter t meets the curve again at parameter -1/t^3.

The equation of a chord with midpoint (h, k): The chord of xy = c^2 with midpoint (h, k) is xk + yh = 2hk (using T = S_1 method).

JEE problems on rectangular hyperbola often involve concyclic points, orthocentre properties, and tangent-normal intersections. The parametric approach (ct, c/t) is almost always the most efficient method.