The problem can be expressed diagrammatically as follows:
This is a parabola of the form $y^2 = 4ax$, as it opens to the right.
It passes through the point $(10,5) \rightarrow 10^2 = 4 \times a \times 5 \rightarrow 100 = 20a \rightarrow a = 5$
Therefore, the focus $= (a,0) = (5,0)$. This is the mid-point of the diameter.