Let $AB$ be the diameter of the parabolic reflector and $V$ its centre.
The perpendicular bisector of $AB$ passes through the centre $V$
Let $V$ be the origin.The perpendicular bisector of $AB$ is the axis of the parabola of which the mirror is a part.
The coordinates of $A$ and $B$ are $(5,10),(5,-10)$
This parabola opens out to the right and the points $AB$ be on it.
Let its equation be $y^2=4ax$
Substituting $(5,10)$ in the equation,we have
Therefore the equation of the parabola is $y^2=20x$
The focus is at $(5,0)$.
Therefore the distance of the focus from the centre is $5$cm.