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Math Help - line point distance

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    line point distance

    Use dot product methods to find the distance from (5,2) to the line 2x-7y=0.
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    Quote Originally Posted by vvc531 View Post
    Use dot product methods to find the distance from (5,2) to the line 2x-7y=0.
    I don't know about dot product methods, but note that the line can be rewritten as y = \frac{2}{7}x.


    That means you're wanting to find the minimum distance between (5, 2) and \left(x, \frac{2}{7}x\right).


    The distance between these points is:

    D = \sqrt{\left(x - 5\right)^2 + \left(\frac{2}{7}x - 2\right)^2}

     = \sqrt{x^2 - 10x + 25 + \frac{4}{49}x^2 - \frac{8}{7}x + 4}

     = \sqrt{\frac{53}{49}x^2 - \frac{78}{7}x + 29}

     = \left(\frac{53}{49}x^2 - \frac{78}{7}x + 29\right)^{\frac{1}{2}}.


    You should now be able to find the minimum distance, by taking the derivative, setting equal to 0 and solving, then using the second derivative test.
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