The function $y=a{x}^2+bx+c$, where $a$ and $c$ are constants, is a quadratic function whose graph in the $xy$-plane is a parabola that opens upward and has a vertex at $(h,k)$. If $k=5$ and the points with $x$-coordinates $10$ and $2$ have the same $y$-coordinate, which of the following must be true?
I. $b<0$
II. $c>5$