Distance from a point to a set

**tarheelborn** · October 11th 2010, 08:00 AM

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.

**Plato** · October 11th 2010, 01:04 PM

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.

**tarheelborn** · October 11th 2010, 01:19 PM

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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