Existence of linear continuous functional on locally convex TVS

Question

Let $c>1$ and $X$ be a locally convex TVS. Take $a,b\in X$ and a closed subspace $Y$ of $X$ such that $Y\cap \{(1-t)a+bt \ | \ t\in\mathbb{R}\}=\emptyset$. How to prove that there exist $f\in X'$ such that $\mathrm{min}\{f(a),f(b)\}=c$ and $f(y)=0$ for every $y\in Y$ ?