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Math Help - Tangent problems

  1. #1
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    Tangent problems

    I'm having trouble with tangent problems, as I often get one tangent right, but the other completely off, or both.

    Here are some examples:

    1) Find the equation of both lines that pass through the origin and are tangent to the parabola y=1+x^2

    y'=2x

    Let a be point of tangency
    m of tangent=(1+a^2-0)/(a-0)
    2a=(1+a^2)/a
    2a^2=1+a^2
    0=a^2-1
    =(a-1)(a+1)

    y=x or y=-x

    Neither of them are tangents, but y=2x and y=-2x are. What did I do wrong?

    2) Find the x-coordinates to the points on the hyperbola xy=1 where the tangents from the point (1,-1) intersect the curve.

    y'=-1/x^2

    Let a be point of tangency

    m of tangent=(1/a+1)/(a-1)
    -1/a^2=a-1/a^2-1
    2/a^2=1
    a^2=2
    a=2^(1/2), -(2^(1/2))

    These are wrong as well.
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  2. #2
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    Your solution a = x = -1,1 are the points of intersection between the line and the parabola therefore they should then be put back into the function x^2+1 giving you two points (-1, 2) and (1,2)

    The two tangent lines can now be found with

    (-1, 2) and (0,0) gives y = -2x

    then

    (1, 2) and (0,0) gives y = 2x

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