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Math Help - Transformation

  1. #1
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    Transformation

    A conic which gived by x^2+xy+y^2=3 Rotated 45 degrees about the origin counterclockwise. What is the quation is transformed to?
    Last edited by Haytham1111; August 30th 2014 at 08:47 AM.
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  2. #2
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    Re: Transformation

    If we rotate the $x$ and $y$ axes (along with our conic) to new axes $x'$ and $y'$, then the old coordinates, in terms of the new ones, are:

    $x = x'\cos 45^{\circ} + y'\sin 45^{\circ} = \dfrac{\sqrt{2}}{2}(x' + y')$

    $y = y'\cos 45^{\circ} - x'\sin 45^{\circ} = \dfrac{\sqrt{2}}{2}(y' - x')$

    So the equation in our new coordinates is:

    $\left( \dfrac{\sqrt{2}}{2}(x' + y')\right)^2 + \left(\dfrac{\sqrt{2}}{2}(x' + y')\right)\left(\dfrac{\sqrt{2}}{2}(y' - x')\right) + \left(\dfrac{\sqrt{2}}{2}(y' - x')\right)^2 = 3$ or:

    $\dfrac{1}{2}(x'^2 + 2x'y' + y'^2) + \dfrac{1}{2}(y'^2 - x'^2) + \dfrac{1}{2}(y'^2 - 2x'y' + x'^2) = 3$, and cancelling and collecting terms:

    $x'^2 + 3y'^2 = 6$

    Compare these two plots:

    x^2+xy+y^2=3 - Wolfram|Alpha

    x^2 + 3y^2 = 6 - Wolfram|Alpha
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  3. #3
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    Re: Transformation

    The answe in the book is 3x^2+y^2=6

    I saw the graph in wolframalpha that is your answer is horisontal transformation, and the boks answer is vertikal one????
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  4. #4
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    Re: Transformation

    Your book is right, I mis-read the question and rotated it CLOCKWISE.

    If we rotate it counter-clockwise, the formula for $x',y'$ are:

    $x = x'\cos 45^{\circ} - y'\sin 45^{\circ} = \dfrac{\sqrt{2}}{2}(x' - y')$

    $y = y'\cos 45^{\circ} + x'\sin 45^{\circ} = \dfrac{\sqrt{2}}{2}(x' + y')$

    which leads to:

    $\left( \dfrac{\sqrt{2}}{2}(x' - y')\right)^2 + \left(\dfrac{\sqrt{2}}{2}(x' = y')\right)\left(\dfrac{\sqrt{2}}{2}(x' + y')\right) + \left(\dfrac{\sqrt{2}}{2}(x' + y')\right)^2 = 3$

    $\dfrac{1}{2}(x'^2 - 2x'y' + y'^2) + \dfrac{1}{2}(x'^2 - y'^2) + \dfrac{1}{2}(x'^2 + 2x'y' + y'^2) = 3$

    $3x'^2 + y'^2 = 6$, as your book says, with this plot:

    3x^2 + y^2 = 6 - Wolfram|Alpha

    I apologize for the mix-up, I "turned it the wrong way"
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  5. #5
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    Re: Transformation

    Hi,
    Deveno's answer is correct. Here's the way I like to think about such problems:



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