The parabolic collector of a solar power plant is 12m wide and 4 m deep. Where should the steam pipe be

The parabolic collector of a solar power plant is 12m wide and 4 m deep. Where should the steam pipe be located (distance from vertex) so that all incident rays will be concentrated on it?

First, Assuming an arbitrary plane perpendicular to the parabola axis - say the plane connecting the edge of the parabolic collector - then the length of the line connecting the plane to a point at the parabola surface plus the length of the line connecting the same point on the surface to the parabola focus will be the same at any point on the parabola surface.

From the above fact the focus of the parabola can be determined hence the location where the steam pipe should be located for max. power.

To find the focus distance from the vertex we assume the following:
a = the perpendicular distance on the axis to the collector edge = 1/2 parabola width = 6m
b = distance from the focus to the collector edge
c = collector depth = 4m
d = distance from the vertex to the focus

Now according to the above stated fact:
(d+c) = (b+0) = sqrt(a^2 + (c-d)^2), squaring both sides,
d^2 + 2dc + c^2 = a^2 + c^2 - 2dc + d^2 , making reduction leads to,
4dc = a^2, so d = (a^2) / 4c = (6^2) / (4*4) = 2.25m

Then the steam pipe should be located 2.25 meters above the vertex and at the center axis of the parabolic collector.