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Math Help - Parabola

  1. #1
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    Parabola

    Hi,
    How do you prove that the projection of the intersection point between two tangents of the parabola projected to the x-axix is at the midpoint between the two points (of tangents with the parabola) on the x axis.?
    Thanks
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  2. #2
    MHF Contributor
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    Hi,

    Let P be the parabola ax^2 + bx + c
    One Cartesian equation of the tangent of the parabola at point x_0, y_0 is
    y = (2a x_0 + b)x - a {x_0}^2 + c

    The abscissa of the intersection point between two tangents of the parabola at points x_0, y_0 and x_1, y_1 is given by

    (2a x_0 + b)x - a {x_0}^2 + c = (2a x_1 + b)x - a {x_1}^2 + c

    Simplification leads to

    2a (x_0 - x_1)x - a (x_0 - x_1)(x_0 + x_1) = 0

    a being non-zero and x0 being different from x1

    x = \frac{x_0 + x_1}{2}
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  3. #3
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    does this work for the parabola y=x^2 as well?
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  4. #4
    MHF Contributor
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    I considered the general equation of parabolas
    At the end the solution is independent with a, b and c
    Therefore it also works for y=x
    Just take a=1, b=0 and c=0
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