Should the radius of the spot on the left-hand diagram not be 3.15m?
Yes, well spotted (now fixed).
So the hyp on the elliptical diagram is 30 metres (centre line of the beam). I can’t see how you calculated the lengths of the two other sides (23.7m and 18.3m) as you have no values for the angles other than the 5.994 half beam to get the longest axis when the light is vertical.
For the above solution the centre line of the beam isn’t the middle of the ellipse. Instead, the lamp is 30m from the centre point of the long axis of the ellipse. As lots of people have said above, the two aren’t the same.The centre line will actually fall to the right of the centre of the long axis of the ellipse (about 30cm further to the right).
I’d suggest you ask DrJ for his solution as it looks very elegant!
Mine was a bit brute force; simultaneous equations involving the height and angle between vertical and centre-line of beam – one based on the triangle of the beam, the second on the right-angled triangle created by the 31.12deg angle.