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Math Help - optimization - distance between points on a graph

  1. #1
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    optimization - distance between points on a graph

    Find the points on the graph of y=9-x^2 that is closest to the point (0, 3).
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  2. #2
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    In general, the distance between two points is

    \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}

    Use:

    (x_1,y_1)=(x,9-x^2) <-- this is a point on the curve
    (x_2,y_2)=(0,3)

    Then we have the distance between a point on the curve and the point (0,3):

    \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}

    =\sqrt{(x-0)^2+(9-x^2-3)^2}

    =\sqrt{x^2+(6-x^2)^2}

    =\sqrt{x^2+36-12x^2+x^4}

    =\sqrt{x^4-11x^2+36}

    Our goal is to find the minimum value of this, so we find the derivative and set it equal to 0 and solve for x.

    Can you take it from here?
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  3. #3
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    Yes, thank you!

    One question though, in the general formula for the distance between two lines, does it matter which function i assign to be (x_1,y_1) and (x_2,y_2)?
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  4. #4
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    Quote Originally Posted by shawli View Post
    Yes, thank you!

    One question though, in the general formula for the distance between two lines, does it matter which function i assign to be (x_1,y_1) and (x_2,y_2)?
    No, it works either way. If you need to be convinced, try switching it around and see that you arrive at the same expression.
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