Let $V$ be an affine algebraic variety defined over $\mathbb R$. We assume that $V\subset \mathbb A_n$(affine $n$ space). Suppose that for any $\mathbb R$-rational map $f:\mathbb P^1\to V$, the image $f (\mathbb P^1(\mathbb R)) $ (here we take the image of points where $f$ can be defined) is contained in a finite union of proper affine $\mathbb R$-subspaces of $\mathbb A_n$. Is it true that $V(\mathbb R)$ is contained in a finite union of affine $\mathbb R $-subspaces of $\mathbb A_n$.

Let $V$ be an affine algebraic variety defined over $\mathbb R$. We assume that $V\subset \mathbb A_n$(affine $n$ space). Suppose that for any algebraic curve $C$ in $V$ defined over $\mathbb R$, the real points $C(\mathbb R)$ is contained in a finite union of proper affine $\mathbb R$-subspaces of $\mathbb A_n$. Is it true that $V(\mathbb R)$ is contained in a finite union of affine $\mathbb R $-subspaces of $\mathbb A_n$.