Step 1:

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.

Step 2:

Let its equation be $y^2=4ax$

Substituting $(5,10)$ in the equation,we have

$100=20a$

$a=5$

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.