From the top of a light house the angles of depression of two ships on the opposite sides of it are observed to be - and - If the height of the light house be h meters and the line joining the ships passes through the foot of the light house- the distance between the ships is

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Standard X

Mathematics

Calculating Heights and Distances

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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 meters and the line joining the ships passes through the foot of the light house, the distance between the ships is

\frac{h (c o t α + c o t β)}{c o t α . c o t β}

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\frac{h (t a n α + t a n β)}{t a n α . t a n β}

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h (t a n α + t a n β)

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\frac{h t a n α . t a n β}{t a n α . t a n β}

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Solution

The correct option is C $\frac{h (t a n α + t a n β)}{t a n α . t a n β}$

Let $A B$ be the lighthouse of height $h$ metres. Let $A C = x$ and $A D = y$ .

In $△ C A B$ ,

$\frac{A B}{A C} = tan α \Rightarrow tan α = \frac{h}{x} \Rightarrow x = \frac{h}{tan α} . . . . . (1)$

In $△ D A B$ ,

$\frac{A B}{A D} = tan β \Rightarrow tan β = \frac{h}{y} \Rightarrow y = \frac{h}{tan β} . . . . . (2)$

Distance between the ships is:

$x + y = \frac{h}{tan α} + \frac{h}{tan β} = h (\frac{1}{tan α} + \frac{1}{tan β}) = h (\frac{tan β}{tan α tan β} + \frac{tan α}{tan α tan β}) = h (\frac{tan α + tan β}{tan α tan β})$ metres

Hence, the distance between the ships is $h (\frac{tan α + tan β}{tan α tan β})$ .

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Similar questions

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 meters and the line joining the ships passes through the foot of the light house, show that the distance between the ship is $\frac{h (t a n α + t a n β)}{t a n α t a n β}$ metres.

Q. 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

\frac{h (\tan α + \tan β)}{\tan α + \tan β}

metres.

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 lighthouse be h metres and the line joining the ships passes through the foot of the lighthouse, show that the distance between the ships is $\frac{h (t a n α + t a n β)}{t a n α t a n β}$ metres. [3 MARKS]