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Math Help - Paramatric equations and LOCUS!!!!

  1. #1
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    Question Paramatric equations and LOCUS!!!!

    p is a variable point on the parabola x^2 = -4y. The tangent from P cuts the parabola x^2 = 4y at Q and R. Show that 3x^2 = 4y is the equation of the locus of the midpoint of the chord RQ.

    I'm clueless.
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  2. #2
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    Let P have co-ordinates (p, -\frac{1}{4}p^2) on the lower parabola.

    The slope of the tangent is -\frac{1}{2}p, and you can see from a sketch that this is equal to

    -\frac{1}{4}p^2 divided by a crucial advance on the x-axis that turns out (as a result) to be

    \frac{1}{2}p.



    And from this you can deduce that the constant in the tangent-line equation is

    \frac{1}{4}p^2.

    So y = -\frac{1}{2}p x + \frac{1}{4}p^2

    is an equation for the tangent line which you can set equal to the upper parabola that it has to (twice) intersect...

    So solve that equation for x, to get the x-values (and after that the y's) of the points of intersection in terms of a, and then finally their mean average (mid-point) in terms of a. Hopefully that final y-value will be three-quarters the square of the x.
    Last edited by tom@ballooncalculus; September 28th 2009 at 12:36 PM. Reason: pic
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  3. #3
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    Thanks but OMG I have to look over that a couple of times to understand it.
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