Aaardwolf´s math is correct if you want to know where the longest line of sight meets the earth surface (assuming earth is sphere).
The problem described in your last post is totally different. If I understood it properly here is a solution (again assuming earth is a sphere):
There are three parallel planes. One goes through the equator, one touches the pole and one goes through a point on the earth surface with a certain horizontal distance from the pole. The drop is the distance between the the latter two planes.

earth radius R
horizontal distance l
drop d

d = R - sqrt(R^2 - l^2)

It´s only valid for l <= R. Posting some graphic may help to understand the problem more clearly.