Let $R$ be an (irreducible) plane real algebraic curve (without isolated points).

Consider Zarissky closure of $R$ in ${\mathbb C}P^1$ (as real variety).

Suppose that $\lambda\in R \Rightarrow\overline{\lambda}^{-1}\in R.$

Question 1: When there exist a polynomial $f(x,y)$ with zero set $R$ such that for each $k\in {\mathbb R}\cup {\infty}$ $f(x,kx)$ has only real roots? (i.e. hyperbolic).

It is possible to state also a weaker version of that question:

Question 2: When there exist a polynomial $f(x,y)$ with zero set $R$ such that for each $k\in {\mathbb R}\cup {\infty}$ holds following property :

If for some $r\in{\mathbb C}$ $f(r,kr)=0$ then $f(\overline{r}^{-1},k\overline{r}^{-1})=0$?