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Math Help - Conics question

  1. #1
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    Conics question

    A straight line D through the focus F of a conic \Phi meets  \Phi in two points A and B.
    Show that the quantity
    \frac{1}{|AF|} + \frac{1}{|BF|}
    is independent of the choice of line D.
    Not sure where to start with this one apart from maybe trying the 3 different conics of hyperbola, parabola and ellipse. Could someone give me a hint where to start?
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  2. #2
    Senior Member
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    All conics can be expressed in polar form as:
    r(\theta) = \frac{l}{1 - e cos(\theta)}, where the focus is at the origin.
    A line that passes through the focus would intersect at points r(\theta_0) and r(\theta_0 + \pi)

    so \frac{1}{|AF|} + \frac{1}{|BF|} = \frac{1}{r(\theta_0)} + \frac{1}{r(\theta_0 + \pi)}

    Can you show that this does not depend on \theta_0?
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