Math CentralQuandaries & Queries


Question from Soumya, a student:

sir,- "Is the graph of an inverse variation a RECTANGULAR HYPERBOLA? If it is, then how can be the equation of a rectangular hyperbola be xy=constant , whereas in the books it is written that the graph of a rectangular hyperbola is- x^2-y^2=a^2?


The two axes of the hyperbola $xy = a$ are the lines $y = x$ and $y = -x;$ its asymptotes are the x- and y-axes.
The two axes of the hyperbola $x^2 - y^2 = b^2$ are the x- and y-axes, while the asymptotes are the lines $y = x$ and $y = -x.$

In other words, if you choose the constants a and b correctly the two curves are identical in every way except for their position; you can rotate either through 45 degrees about the origin to get the other. To do this algebraically, you can change coordinates by replacing $x$ in the first equation by $(u - v)\sqrt 2$ and $y$ by $(u + v)\sqrt 2.$ Then $xy = a$ becomes

\[(u - v)\sqrt 2 \times (u + v)\sqrt 2 = (u^2 - v^2)/2 = a.\]

This tells us that if we choose $2a = b^2,$ our two hyperbolas are congruent -- that is, you can rotate one into the other.


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