Question

The equation of the rectangular hyperbola with its asymptotes as the coordinate axes is xy = c² , where c is constant, is called the simplest form of a rectangular hyperbola.

Answer 1

Let x² - y² = a² be rectangular hyperbola. Now, rotate the axes about the origin through π/4 in the clockwise direction so that the asymptotes become coordinate axes [because the angle between the asymptotes of x² - y² = a² is 2sec^-1 (√2) which equal to  π/2 ]. Then

We write c = a/√2 so that xy = c² . Also xy = c² ⇒either both x and y are positive or both x and y are negative.