Question

From an aeroplane vertically above a straight horizontal road, the angles of depression of two consecutive milestone on opposite sides of the aeroplane are observed to be α, and β. Show that the height in miles of aeroplane above the road is `(tanα  tanβ)/(tanα + tanβ)`.

Answer 1

Let P Q be h
QB be x
Given : AB = 1 mile
QB = x
AQ = (1-x) mile
in ΔPAQ
`Tan  α = "PQ"/"AQ"`

`Tan  α = "h"/(1-"x")`

`1 - "x" = "h"/(Tan  α)`      ............1

In ΔPQB

`Tan β = "h"/"x"`

`"x" = "h"/(Tan  β)`

Substitute for x in equation (1)

`1 = "h"/Tan β + "h"/(Tan  α)`

`1 = "h"{1/Tan β + 1/Tan α}`

`1/"h" = (Tan β + Tan α)/(Tan β Tan α)`

Thus , the height in miles of aeroplane above the road is `(Tan α Tan β)/(Tan α + Tan β)`