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Math Help - Distance from a point to a set

  1. #1
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    Distance from a point to a set

    I need to find (in R^2) \rho((4,2), A_2) where A_2=\{(x,y)|x^2+y^2=1 \} .

    I know that the definition says that this distance is the greatest lower bound of  \{\rho(x,a)|a \in A\}. So the distance between \rho((4,2), (x,y))= \sqrt{(x-4)^2+(y-2)^2}=\sqrt{x^2+8x+16+y^2-4y+4}= \sqrt{21-8x-4y} and then I am not sure what to do.
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  2. #2
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    The easy way is to write the equation of the line through (0,0)~\&~(4,2).
    Find the its intersection with the circle.
    One of those two points is the closest.
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  3. #3
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    Thanks so much! I had made this problem quite hard by trying to use polar coordinates, but this method is much simpler. Thank you so much!
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