algebra

Please help with this problem:
An ellipse and a hyperbola have the same foci, $A$ and $B$, and intersect at four points. The ellipse has major axis 50, and minor axis 40. The hyperbola has conjugate axis of length 20. Let $P$ be a point on both the hyperbola and ellipse. What is $PA \cdot PB$?

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  1. If you set up the curves as described, you have

    x^2/25^2 + y^2/20^2 = 1
    x^2/125 - y^2/100 = 1

    They intersect at (±25√5/3,±40/3) so pick P to be one of those points and figure your answer.

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  2. Steve means the intersections are (±(25√5)/3,±(40)/3) if anyone is confused.
    Focus 1 (A): (15, 0)
    Focus 2 (B): (-15, 0)
    P: ((25√5)/3,(40)/3)
    The question is asking for the distance between P and A, and multiply that by the distance between P and B.
    Distance between P and A: √(750 - 250√5)
    Distance between P and B: √(750 + 250√5)
    Our final answer is 500, which is correct.

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