Given this hyperbola $x^2 - y^2 = 1$, I want to rotate it by $45°$ (so the curve is only in the 1st and 3rd quadrant). How to write the formula of my new hyperbola. And can I find which rational function this new rotated hyperbola represents?
I know there is a way to rotate hyperbolas with matrices, but I would like to avoid that, since we haven't discussed the topic yet in class.
Observe that with the 45-degree transformed coordinates $x=\frac{1}{\sqrt2}(u+v),\>\>\>\>\>y=\frac{1}{\sqrt2}(u-v)$. the equation becomes, $2uv = 1$.
Thus, after a 45-degree rotation, the hyperbola $x^2 - y^2 = 1$ become,
$$xy = \frac12$$