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Math Help - Orthogonal trajectories?

  1. #1
    Newbie
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    Sep 2008
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    Orthogonal trajectories?

    How do I show two equations are orthogonal?
    x+y=0
    x=sin y

    and

    How do I verify that the two families of curves are orthogonal when K and C are real numbers?
    xy= C
    x^2-y^2=K

    These questions popped up on my hw and I'm kinda lost about how to do them.
    Thanks in advance!
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  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
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    Well, they're orthogonal at a point if y'_1(p)=-\frac{1}{y'_2(p)} so for the first I'd write them as y_1=-x and y_2=\arcsin(x). Calculate derivatives and show at the origin the orthogonal requirement is met. For the second one:

    2x-2y_2\frac{dy_2}{dx}=0

    \frac{dy_2}{dx}=\frac{x}{y}\Rightarrow \frac{dy_1}{dx}=-\frac{y}{x}

    or:

    y_1=\frac{C}{x}
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