Question

From the top of a lighthouse, the angles of depression of two ships on the opposite sides of it are observed to be α, and β. If the height of the light house is 'h' m and the line joining the ships passes through the foot of the light house, show that the distance between the ship is `("h"(tan α + tan β))/(tanα  tanβ)`m. 

Answer 1

Let AB be the lighthouse of height h m. Let AC = x and AD = y .

In ΔCAB,

`"AB"/"AC" = tan  α`

`tan α = "h"/"x"`

`"x" = "h"/tanα`  ....(i)

In ΔDAB,

`"AB"/"AD" = tanβ`

`tanα = "h"/"y"`

`y = "h"/tanβ` ....(ii)

Distance between the ships = x + y

= `"h"/tanα + "h"/tanβ`

= `"h"((tanβ + tanα)/(tanα  tanβ))`