Distance from a point to a set

1. The problem statement, all variables and given/known data
I need to find [tex] \rho((4,2), A_2)[/tex] where [tex]A_2=\{(x,y)|x^2+y^2=1 \} [/tex].

2. Relevant equations



3. The attempt at a solution
I know that the definition says that this distance is the greatest lower bound of [tex] \{\rho(x,a)|a \in A\}[/tex]. So the distance between [tex]\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}[/tex] and then I am not sure what to do.
 

HallsofIvy

Science Advisor
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1. The problem statement, all variables and given/known data
I need to find [tex] \rho((4,2), A_2)[/tex] where [tex]A_2=\{(x,y)|x^2+y^2=1 \} [/tex].

2. Relevant equations


3. The attempt at a solution
I know that the definition says that this distance is the greatest lower bound of [tex] \{\rho(x,a)|a \in A\}[/tex]. So the distance between [tex]\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}[/tex] and then I am not sure what to do.
Well, that is [itex]D= (x^2+ y^2+ 8x- 4y+ 20)^{1/2}[/itex]. Differentiate that and set the derivative equal to 0 to find the minimum.

Or, use geometry! [itex]x^2+ y^2= 1[/itex] is a circle with center at the origin and radius 1. The nearest point to (4, 2) on that circle lies on the straight line from (0, 0) to (4, 2) (an extended radius). The equation of the line through (0, 0) and (4, 2) is y= x/2. Where does that line intersect the circle [itex]x^2+ y^2= 1[/itex]
 

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