# Rotation of a hyperbola in affine geometry

Given the hyperbola $x^2 - 3xy + y^2 + 4x - 5y + 2 =0$
I have translated this by $x+\frac{7}{5}$ and $y-\frac{2}{5}$ and got
$x^2 - 3xy + y^2 = \frac{9}{5}$

Now, the bit where I'm stuck; I have rotated by $\frac{\pi}{4}$ and got

$5x^2 - y^2 = 18/5$

but need to put this in standard form, which for a hyperbola would be $xy=1$?

But I really don't know where to begin. By plotting the equation it looks as if it needs another rotation to get to $xy=1$ but I am not sure. The book I am using seems to suggest it requires another affine transformation but not necessarily a rotation.

Try scaling the $x$-axis with $x\mapsto \frac{1}{\sqrt{5}}x$ and then rotating by $-\frac{\pi}{4}$.