A Rotated Reciprocal Hyperbola

Recall that the graph of an inverse-proportion function $y=\dfrac kx$ ($k\neq 0$) is a hyperbola whose two asymptotes are the coordinate axes.

1. Find the length of the transverse axis of the graph $C_0$ of $y=\dfrac 1{2x}$. 2. Rotate the curve $C_0$ clockwise by $\dfrac{\pi}4$ about the origin to obtain a curve $C$. (a) Find the equation of $C$. (b) Let $A$ be the left vertex of $C$. A circle $E:(x-1)^2+(y-1)^2=r^2$ ($r>0$) meets the line $l:x=1$ at points $P$ and $Q$; the lines $AP$ and $AQ$ meet the hyperbola $C$ again at points $M$ and $N$. Determine whether the distance from $A$ to the line $MN$ has a maximum value. If so, find the maximum and the corresponding value of $r$; if not, explain why.