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Math Help - xy(x^2 - y^2) = C

  1. #1
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    xy(x^2 - y^2) = C

    I wanted to plot xy(x^2-y^2)=C for C=840 but realised I couldn't isolate either x or y
    I went to quickmath.com and entered the general equation and got...

    Solve Equation :: QuickMath.com - Automatic Math Solutions

    This returned 3 equations for calculating y; 2 complex and 1 real.

    I know that for xy(x^2-y^2)=840 (7, 3), (7, 5) and (8, 7) are co-ordinates on the curve.
    I cannot get these results by plugging in C=840, x=7 to the equations returned by quickmath.com.

    Am I doing something wrong?

    Thx

    Ben
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  2. #2
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    Re: xy(x^2 - y^2) = C

    Hi Ben.

    That is certainly a messy equation when you try to rearrange for y. Personally I would change coordinates.
    Changing to polar:  x = r \cos \theta \,\ y = r \sin \theta , we get.

     r \cos \theta \cdot r \sin \theta \left( r^2 \cos^2 \theta - r^2 \sin^2 \theta \right) = C

     \frac{1}{2} r^4 \sin 2 \theta \cos 2 \theta  = C

     \frac{r^4 }{4} \sin 4 \theta = C \,\ \rightarrow \,\ r^4 \sin 4 \theta = 4C

    Maybe it's easier to work with from here.
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  3. #3
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    Re: xy(x^2 - y^2) = C

    Quote Originally Posted by pomp View Post
    Hi Ben.

    Maybe it's easier to work with from here.
    Thx pomp, yes, it's easier to do it this way from a graphing point of view.
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  4. #4
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    Re: xy(x^2 - y^2) = C

    The way you have it, 12 solutions:
    x , y
    -8,-7
    -7,-5
    -7,-3
    -7,8
    -5,7
    -3,7
    3,-7
    5,-7
    7,-8
    7,3
    7,5
    8,7
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  5. #5
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    Re: xy(x^2 - y^2) = C

    That's right Wilmer, the equation produces 4 hyperbolae, one in each quadrant. My interest is in the top-right quadrant. i.e. where x and y are both positive.
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  6. #6
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    Re: xy(x^2 - y^2) = C

    Capish!
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