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Math Help - Orthogonal curves?

  1. #1
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    Orthogonal curves?

    This is just crazy. My prof never went over this stuff and I have been searching on how to do these. I don't even know how to plot them. Please help!

    Are the curves orthogonal:
    x^2+y^2=k where k>0
    xy=c where c does not = 0.

    Thanks so much for the help!
    Last edited by mr fantastic; April 12th 2009 at 03:21 PM. Reason: Modified language
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  2. #2
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    Quote Originally Posted by starbuck View Post
    This is just crazy. My prof never went over this stuff and I have been searching on how to do these. I don't even know how to plot them. Please help!

    Are the curves orthogonal:
    x^2+y^2=k where k>0
    xy=c where c does not = 0.

    Thanks so much for the help!
    Two families of curves are orthogonal if their derivatives are negative reciprocals of each other. Using implict differentation we get


    2x+2y\frac{dy}{dx}=0 \iff \frac{dy}{dx}=-\frac{x}{y}

    and

    (1)y+x\frac{dy}{dx}=0 \iff \frac{dy}{dx}=-\frac{y}{x}

    So the answer is NO.

    Just for fun we can solve for the orthogonal family to the circle

    2x+2y\frac{dy}{dx}=0 \iff \frac{dy}{dx}=-\frac{x}{y}

    So taking the negative reciprocal of the slope we get

    \frac{dy}{dx}=\frac{y}{x} \iff \frac{dy}{y}=\frac{dx}{x}

    Integrating both sides we get

     \ln|y|=\ln|x|+c \iff y=e^cx \iff y=mx

    So the orthogonal family is the set on lines that pass through the point (0,0)
    Last edited by mr fantastic; April 12th 2009 at 03:21 PM.
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  3. #3
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    Thanks, I just have a quick question. Why did you set the second one (xy=C) to zero when C could not equal 0? Can you still draw trajectories for this problem, or is that only if they are orthogonal?
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    Behold, the power of SARDINES!
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    Quote Originally Posted by starbuck View Post
    Thanks, I just have a quick question. Why did you set the second one (xy=C) to zero when C could not equal 0? Can you still draw trajectories for this problem, or is that only if they are orthogonal?

    The derivative of a constant is 0 for any constant c zero or not. You could plot them but the would not be orthogonal.
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    Ahhh, okay. Thanks soooooooooooo much.
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