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Math Help - [SOLVED] Error in this tangent line problem?

  1. #1
    Member sinewave85's Avatar
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    [SOLVED] Error in this tangent line problem?

    This is from a review section, and I am wondering whether there is a typo in the original problem itself or if I have forgotten something critical. Here is the problem:

    "Find the point(s) on the graph of f(x) = -x^2 such that the tangent line at that point passes through the point (0,-9)."

    Just thinking about the graph of f(x) = -x^2, it would seem impossible for there to be a tangent which passes throught (0,-9). But just for good measure, I did the math:

    \mbox{Let } f(x) = f(c) \mbox{ and } (x,y) = (c,-c^2)

    m = f^{\prime}(c) = -2c

    m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

    m = \frac{-9 + c^2}{0 - c} = \frac{-9 + c^2}{-c}

    -2c = \frac{-9 + c}{-c}

    2c^2 = -9 + c^2

    c^2 = -9

    c = \sqrt{-9}

    Sure enough, there appears to be no real answer. Am I making a mistake, or is this a question without an answer?
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by sinewave85 View Post
    This is from a review section, and I am wondering whether there is a typo in the original problem itself or if I have forgotten something critical. Here is the problem:

    "Find the point(s) on the graph of f(x) = -x^2 such that the tangent line at that point passes through the point (0,-9)."

    Just thinking about the graph of f(x) = -x^2, it would seem impossible for there to be a tangent which passes throught (0,-9). But just for good measure, I did the math:

    \mbox{Let } f(x) = f(c) \mbox{ and } (x,y) = (c,-c^2)

    m = f^{\prime}(c) = -2c

    m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

    m = \frac{-9 + c^2}{0 - c} = \frac{-9 + c^2}{-c}

    -2c = \frac{-9 + c}{-c}

    2c^2 = -9 + c^2

    c^2 = -9

    c = \sqrt{-9}

    Sure enough, there appears to be no real answer. Am I making a mistake, or is this a question without an answer?
    I think you're right. The point (0,-9) lies inside the parabola, so therefore there can be no tangent line to the parabola passing through this point.
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  3. #3
    Member sinewave85's Avatar
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    Quote Originally Posted by redsoxfan325 View Post
    I think you're right. The point (0,-9) lies inside the parabola, so therefore there can be no tangent line to the parabola passing through this point.
    Thanks for the reinforcement, redsoxfan325. I guess I will just submit the problem as I have it worked up here. There is the possibility that the problem was purposefully designed to have no answer, but I think that it is more likely that the function and the point were supposed to have opposite signs.
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