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Prove some properties of conic sections

  1. Consider the Cartesian equation

        \[ \frac{x^2}{a^2} + \frac{y^2}{a^2 - c^2} = 1. \]

    Prove that this equation represents all conic sections symmetric about the origin with foci at (c,0) and (-c,0).

  2. Let c be a fixed constant and let S be the set of all such conics as a^2 takes on all positive values other than c^2. Prove that every curve in the set S satisfies the differential equation

        \[ xy \left( \frac{dx}{dy} \right)^2 + (x^2 - y^2 - c^2) \frac{dy}{dx} - xy = 0. \]

  3. Prove that the set S is self-orthogonal. This means that the set of all orthogonal trajectories of curve in S is the set S itself.


One comment

  1. S says:

    a) follows from the theorem 13.19.
    b) expressing the 3 terms of the differential equation on the left with the conic section equation (derivative), and summing them up, leads to 0. That means that the differential equation holds, but hopefully there is a smarter way to derive it.
    c) just plugging in the hint given in the book shows that the same differential equation holds for the orthogonal trajectory. However, I think ideally we should show that each integral curve of the differential equation is the conic section, but I didn’t do that.

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