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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.


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