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Math Help - derivative application

  1. #1
    Junior Member
    Joined
    Jan 2009
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    derivative application

    Find the equation of any quadratics that passes through the origin and are tangent to both y=-2x-4 and y=8x-49.

    I have thought about this problem for most of the afternoon now and got no where. I tried drawing it to see the problem but no good. I just don't know where to start or how to approach this problem. What/how should I be thinking?

    Thank you
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  2. #2
    MHF Contributor
    Joined
    Nov 2008
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    France
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    1,458
    Hi

    The quadratics you are looking for passing through the origin, their generic equation is f(x) = ax^2 + bx

    The tangent at a point whose abscissa is x_0 is :
    T_0 : y = (2ax_0+b)x-ax_0^2

    The quadratics are tangent to both y=-2x-4 and y=8x-49 iff there exists x_0 and x_1 such that
    2ax_0+b = -2
    -ax_0^2 = -4
    2ax_1+b = 8
    -ax_1^2 = -49

    Solve for a, b, x0 and x1.
    You will find 2 quadratics.
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