text: # Thread: Distance from a point to a set
  
  1. ## Distance from a point to a set
  
  I need to find $\displaystyle (in R^2) \rho((4,2), A_2)$ where $\displaystyle A_2=\{(x,y)|x^2+y^2=1 \}$.
  
  I know that the definition says that this distance is the greatest lower bound of $\displaystyle \{\rho(x,a)|a \in A\}$. So the distance between $\displaystyle \rho((4,2), (x,y))= \sqrt{(x-4)^2+(y-2)^2}=\sqrt{x^2+8x+16+y^2-4y+4}= \sqrt{21-8x-4y}$ and then I am not sure what to do.
  
  2. The easy way is to write the equation of the line through $\displaystyle (0,0)~\&~(4,2)$.
  Find the its intersection with the circle.
  One of those two points is the closest.
  
  3. 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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