$A$ and $B$ are real $n\times n$ matrices and $S=\{(x,y)\in \mathbb{R^2}$ : $I+xA+yB$ is positive semidefinite matrix $\}$.
Prove that $\mathbb{R^2}= Aff(S)$ if and only if $A$ and $B$ are symmetric. Here $Aff(S)$ stands for affine span of $S$. I have proved that if $\mathbb{R^2}= Aff(S)$ then $A$ and $B$ are symmetric. I have problem with proving other direction. Any ideas would be helpful.
Since $S \subset \mathbb{R^2}$ then $Aff(S) \subset \mathbb{R^2}$. It is obvious since $\mathbb{R^2}$ is affine subspace. I am not sure how to prove other inclusion if it is known that matrices $A$ and $B$ are symmetric.
Hint: It's enough to show $(0,0),\ (x,0),\ (0,y)\in S$ for some nonzero $x, y$.
Then find such an $x$ using the eigenvalues of $A$.