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Math Help - Orthogonal Trajectories

  1. #1
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    Orthogonal Trajectories

    Hi, how do I find the orthogonal trajectories of the family of ellipses:

    \frac{x^2}{a^2} + y^2 = k^2

    where a is a constant?

    Thanks.
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  2. #2
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    Quote Originally Posted by coldfire View Post
    Hi, how do I find the orthogonal trajectories of the family of ellipses:

    \frac{x^2}{a^2} + y^2 = k^2

    where a is a constant?

    Thanks.
    First calculate the slope of the tangent

     <br />
\frac{2x}{a^2} + 2 y \frac{dy}{dx} = 0 \;\text{or}\; \frac{dy}{dx} = - \frac{x}{a^2 y}.<br />

    Orthogonal trajectories will be perpendicular so

    \frac{dy}{dx} = \frac{a^2y}{x}.

    Now integrate.
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