# Thread: Rotating a hyperbola: standard form

1. ## Rotating a hyperbola: standard form

Hello, I am confused by an equation of a hyperbola, xy=1. It doesn't fit the standard form x^2/a^2 - y^2/b^2 = 1. I know that it can be rotated 45 degrees to get it into standard form, but i don't know how to start.. Thank you for your help in advance

2. ## Re: Rotating a hyperbola: standard form

let's just look at standard form

$x y = 1$

we want to rotate this 45 degrees to align it with

$\dfrac{x^2}{a^2} - \dfrac{y^2}{a^2} = 1$

the rotation matrix is given by

$\dfrac {1}{\sqrt{2}}\begin{pmatrix}1 &-1 \\1 &1 \end{pmatrix}$

$\begin{pmatrix}x \\ y \end{pmatrix} \Rightarrow \begin{pmatrix}\dfrac {1}{\sqrt{2}} (x-y) \\\dfrac {1}{\sqrt{2}}(x+y) \end{pmatrix}$

substituting these in we get

$\left(\dfrac {1}{\sqrt{2}} (x-y)\right)\left(\dfrac {1}{\sqrt{2}}(x+y)\right) = 1$

$\dfrac 1 2 (x^2 - y^2) = 1$

$\dfrac{x^2}{2} - \dfrac{y^2}{2} = 1$

$a = \sqrt{2}$