From the top of a lighthouse, the angles of depression of two ships on the opposite sides of it are observed to be a and 3. If the height of the lighthouse be h meters and the line joining the ships passes through the foot of the lighthouse, show that the distance

`(h(tan alpha + tan beta))/(tan alpha tan beta)` meters

#### Solution

Let h be the height of lighthouse AC. And an angle of depression of the top of the lighthouse from two ships is α and β respectively.

Let BC = x, CD = y. And ∠ABC = α, ∠ADC = β.

We have to find the distance between the ships

We have the corresponding figure as follows

We use trigonometric ratios.

In ΔABC

`=> tan α = (AC)/(BC)`

`=> tan α = h/x`

Again in Δ ADC

`=> tan β = (AC)/(CD)`

`=> tan β = h/y`

`=> y = h/(tan β)`

Now

`=> BD = x + y`

`=> BD = h/(tan α) + h/(tan β)`

`=> BD = (h(tan α + tan β))/(tan α tan β)`

Hence the distance between ships is `(h(tan α + tan β))/(tan α tan β)`