A hyperbola is said to be a rectangular hyperbola if its asymptotes are at right angles.
The angle between the asymptotes is given by 2 tan−1 (b/a).
b/a = tan 45° = 1
a = b
The equation of rectangular hyperbola is
x2 − y2 = a2
The combined equation of the asymptotes is x2 − y2 = 0. The separate equations are x − y = 0 and x + y = 0. i.e., x = y and x = −y. The transverse axis is y = 0, conjugate axis is x = 0.