I'm trying to find a satisfying answer to this problem, so I would appreciate some help.
Let $p\in S$ be an elliptic point and let $r$ and $r'$ be conjugate directions at $p$. Varying $r$ in $T_pS$, show that the minimum value of the angle between $r$ and $r'$ is attained by a unique pair of vectors in $T_pS$, which are symmetric with respect to the principal directions.
My idea was to simply consider unit vectors in directions of $r$ and $r'$ respectively, say $w=\cos(\theta)e_1+\sin(\theta)e_2$ and $w'=\cos(\phi)e_1+\sin(\phi)e_2$ so that the angle between these two vectors would be given by $\displaystyle \cos(\theta)\cos(\phi)+\sin(\theta)\sin(\phi)$.
Now, remember that $\theta$ is varying and, since $r$ and $r'$ are conjugate, $\phi=\phi(\theta)$. Taking the derivative of that last expression with respect to $\theta$ and making it equal to 0, we should be able to find the answer.
I'm not very confident of this approach, though.
Thank you!