A flagstaff stands on the top of a vertical tower. The angle of elevation of the top of the flagstaff from a certain place on the same horizontal level with the base of the tower is found to be α. Advancing a distance d towards the tower in the same horizontal plane, the angle of elevation of the top of the flagstaff is observed to be β and that of the top of the tower is observed to be γ. Let H be the height of the top of the flagstaff from the base of the tower and h be the height of the tower. Which one of the following is correct?
- AH tan γ − h tan β = 0
- Bh tan γ − H tan β = 0
- CH tan γ − h tan α = 0
- Dh tan γ − H tan α = 0
Explanation
Let the tower have height h and the flagstaff height f, so the top of the flagstaff is H = h + f above the base. After advancing distance d, let the observer be x₂ from the tower base: then tanβ = H/x₂ (top of flagstaff) and tanγ = h/x₂ (top of tower). Both share the same adjacent side x₂, so x₂ = H/tanβ = h/tanγ, giving H·tanγ = h·tanβ, i.e. H tanγ − h tanβ = 0. Only option (A) states this relation.