What is the shortest curve $\gamma$ in $\mathbb{R}^2$ from the origin $o=(0,0)$ to a rational point $p=(a,b)$ that (a) passes through no other rational point, and (b) contains no point a rational distance from both $o$ and $p$?

A rational point is one with rational coordinates. I am wondering if one can reach $p$ via such a highly "irrational route." My guess is that there are curves whose lengths approach the straight-line distance $|p|$. Perhaps the curve should be restricted to a specific class: $C^\infty$, analytic, elliptic, quadratic, circle arc. (Just avoiding rational points [condition (a)] can be accomplished with a circle arc of an appropriately chosen radius.)

This is far from my experience, and it may be that all the variations have trivial, uninteresting answers, in which case I apologize for the distraction.