Question

From the top of a light house, 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 be h metres and the line joining the ships passes through the foot of the light house, show that the distance between the ship is 

`(h (tan ∝+tan ß))/ (tan ∝+tan ∝)`

Answer 1

Let h be the height of light house AC. And an angle of depression of the top of light house from two ships are ∝ and ß respectively. Let` BC=x, CD=y`,. And ∠ABC =∝, ∠ADC=ß , .

We have to find distance between the ships

We have the corresponding figure as follows 

We use trigonometric ratios.

In `Δ ABC` 

⇒` tan ∝ =(AC)/(BC)` 

⇒` tan ∝= h/x` 

⇒` x= h/ tan ∝`

Again in `Δ ADC` 

⇒ `tan ß=(AC)/(CD)` 

⇒` tan ß= h/y`

⇒` y=h/tan ß`

Now, 

⇒ `BD=x+y` 

⇒ `BD= h/tan ∝ +h/tan ß` 

⇒ `BD= (h(tan∝+tan ß ))/(tan ∝ tanß)` 

Hence the distance between ships is `(h(tan∝+tan ß))/(tan∝ tan  ß)`