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Thread: Rotating a hyperbola: standard form

  1. #1
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    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
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  2. #2
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    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

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

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

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

    $a = \sqrt{2}$
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