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

Consider Zarissky closure of $R$ in ${\mathbb C}P^1.$

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

Question: 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).