rotation of hyperbola How can we rotate the rectangular hyperbola
xy=c   ( c is any constant) 
Into a form of standard hyperbola that is
(x/a)$^2$ - (y/b)$^2$ = 1
By rotating the hyperbola . 

$xy = c$
let $x = u-v\\ y = u+v$
$xy = u^2 - v^2 = c$
Now the transformation that I have just done has a little bit of spacial compression to it.  
If you do a traformation along the lines of $x = au-bv, y = bu+av$ then there will be compression on the order of  $\sqrt {a^2 + b^2}$  So, it is not a bad idea to choose $a,b$ such that $a^2 + b^2 = 1$
or $x = \cos \phi u - \sin\phi v\\ y = \sin\phi u + \cos\phi v$
and by trig identity that $cos^2 \phi + sin^2 \phi = 1$
$x = \frac {\sqrt 2}{2} u-\frac {\sqrt 2}{2}v\\ y = \frac {\sqrt 2}{2}u+\frac {\sqrt 2}{2}v\\
xy = \frac {u^2}2 + \frac {v^2}2 = c\\
 \frac {u^2}{2c} + \frac {v^2}{2c} = 1 $
