



 
Soumya, The two axes of the hyperbola $xy = a$ are the lines $y = x$ and $y = x;$ its asymptotes are the x and yaxes. 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. Chris  


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