MathDB

Consider the following subsets of the plane:

C_1=\Big\{(x,y)~:~x>0~,~y=\frac1x\Big\}

and

C_2=\Big\{(x,y)~:~x<0~,~y=-1+\frac1x\Big\}

Given any two points

P=(x,y)

and

Q=(u,v)

of the plane, their distance

d(P,Q)

is defined by

d(P,Q)=\sqrt{(x-u)^2+(y-v)^2}

Show that there exists a unique choice of points

P_0\in C_1

and

Q_0\in C_2

such that d(P_0,Q_0)\leqslant d(P,Q) \forall ~P\in C_1~\text{and}~Q\in C_2.

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