The rectangular hyperbola xy = $c^2$ is the most common non-standard form in JEE. It is obtained by rotating $x^2$ - $y^2$ = 2$c^2$ 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): $\frac{x}{ct}$ + $\frac{y}{c/t}$ = 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 + y*$t_1$*$t_2$ = 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_1$$t_2$$t_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 (-$\frac{c}{t_1*t_2*t_3}$, -c*$t_1$$t_2$$t_3$). If the fourth point of the concyclic set is $t_4$ = $\frac{1}{t_1*t_2*t_3}$, 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.