If the lengths of transverse and cojugate axes of any hyperbola be equal then it is called rectangular or equilateral hyperbola.

Since length of transverse axis and conjugate axis are same i.e. `a= b`

then, ` x^2/a^2-y^2/b^2=1` becomes `x^2 - y^2 = a^2`

Eccentricity, `e =sqrt((1+b^2/a^2))=sqrt2`

All the results of `x^2/a^2-y^2/b^2=1` are applicable to the hyperbola `x^2 - y^2 =a^2` after changing `b` by `a`.

The Rectangular Hyperbola `xy = c^2`:

When the centre of any rectangular hyperbola be at the origin and its asymptotes coincide with the co-ordinate axes then equation of hyperbola is `xy = c^2`
Here the equation of asymptotes is `xy = 0` and the equation conjugate hyperbola is `xy = -c^2` .


Properties of Rectangular Hyperbola `xy=c^2`:

(a) Equation of a chord joining the points `P(t_1)` & `Q(t_2)` is `x + t_1t_2y = c(t_1 +t_2)` with slope `m =-1/(t_1t_2)`.

(b) Equation of the tangent at `P(x_1,y_1)` is `x/x_1+y/y_1=2` & at `P(t)` is `x/t+ty=2c`.

(c) Point of intersection of tangents at `'t_1'` and `'t_2'` is `((2ct_1t_2)/(t_1+t_2), (2c)/(t_1+t_2))`.

(d) Equation of normal is `y -c/t=t^2(x-ct)` or `xt^3-yt-ct^4+c=0`

(e) Equation of normal `(x_1,y_1)` is `x x_1- yy_1 = x_1^2- y_1^2`.

(f) Chord with a given middle point as `(h, k)` is `k x + hy = 2hk` .