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Question
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)`
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