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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 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, find the distance between the ships is metres.

A
h(tan α+tan β)tan α tan β
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B
h(sec α+tan β)tan α tan β
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C
h(tan α+cot β)tan α tan β
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D
h(cosec α+tan β)tan α tan β
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Solution

## The correct option is A h(tan α+tan β)tan α tan βLet AB be the lighthouse and C and D be the positions of the two ships. Then, AB = h metres. Clearly, ∠ACB=α and ∠ADB=β Let AC = x metres and AD = y metres From right ΔCAB, we have ACAB=cot α⇒xh=cot α⇒x=h cot α ........... (i) From right ΔDAB, we have ADAB=cot β⇒yh=cot β⇒y=h cot β ........... (ii) Adding the corresponding sides of (i) and (ii), we get x+y=h(cot α+cot β)=h(1tan α+1tan β) ⇒x+y=h(tan α+tan β)tan α tan β Hence, the distance between the ships is h(tan α+tan β)tan α tan β m

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