From the top of a tower h meter high, the angle of depression of two objects, which are in line to the foot of the tower is α and β (β>α). Find the distance between the two objects.
Answer:
(cotα−cotβ)h meters
- Let AB be the tower of height h meter and x meter be the distance between the two objects C and D.
As β>α, β will be the angle of depression of the point D and α will be the angle of depression of the point C.
The situation given in the question is represented by the image given below.
- In the right-angled triangle ABD, we have tanβ=ABAD⟹tanβ=hAD⟹AD=htanβ⟹AD=h cotβ[As,cotβ=1tanβ]…(i)
- In right-angled triangle ABC, we have tanα=ABAC⟹tanα=hAC⟹AC=htanα⟹AD+x=h cotα[ As,cotα=1tanα and AC = AD + x ]…(ii)
- Now, let us subtract eq (i) from eq (ii). (AD+x)−AD=h cotα−h cotβ⟹x=(cotα−cotβ)h
- Therefore, the distance between two objects is (cotα−cotβ)h meters.