I have been given the following question on a homework sheet:
I was wondering if I could have some help. I have this so far:
$x^2-y^2+2xy-1 = (x+y)^2-2y^2-1 = 0$
and so $(x+y)^2-2y^2=1$.
I can clearly see that this is a hyperbola. But if I let
$x'=x+y$ and
$y'=y$
so that it is in the correct form, I believe this actually distorts the curve.
How can I make a suitable change of co-ordinates to show that it is a hyperbola without distorting the curve? My thoughts are along the same lines as normalisation and orthogonal bases, but I am just looking for some guidance as to whether I am heading in the correct direction and a way to do so without distorting the curve.
