Let C be a smooth conic inside a projective variety $X$ with ample class $H$. Since $C$ is smooth we have $C \cong \mathbb{P}^1$ and $\mathcal{O}(H)|_C \cong \mathcal{O}_C(2)$, where I think we interpret $\mathcal{O}_C(2)$ as $\mathcal{O}_{\mathbb{P}^1}(2)$. Let $L$ be a "line" in $X$, i.e. a subscheme of $X$ with Hilbert polynomial $p_L(t)=1+t$. Note that $C$ will have Hilbert polynomial $p_C(t)=1+2t$.
My question is: do we have $C \sim_{\mathrm{rat}} L$ or $C \sim_{\mathrm{rat}} 2L $, where $\sim_{\mathrm{rat}}$ denotes rational equivalence (in the Chow ring)?
Since $C \cong \mathbb{P}^1$ and $\mathbb{P}^1$ is a line, is seems like we would have the former case for a smooth conic? But e.g. for the case of a reducible conic $C = L_1 \cup L_2$ where we have two distinct lines intersecting at a point, it seems like we would have $C \sim_{\mathrm{rat}} 2L$? Am I correct in both cases here, or should they both (smooth and reducible) rationally equivalent to $2L$?
Thanks in advance for any help.