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Question
If the angle of elevation of a cloud from a point h meters above a lake is a*and the angle of depression of its reflection in the lake is |i. Prove that the height of the cloud is `(h (tan β + tan α))/(tan β - tan α)`.
Solution
Let LM be the upper surface of the lake and A be a point such that AL = h.
Let C be the position of the cloud and C' be its reflection in the lake.
CM = MC' = x(let)
∠BAC = α and ∠BAC' = β
Now In ΔCBA,
tan α = `"CB"/"AB"`
tan α = `(x - h)/"AB"`
AB = `(x - h)/(tan α)` .....(i)
In ΔC'BA,
tan β = `"CB"/"AB"`
tan β = `(x + h)/"AB"`
AB = `(x + h)/tan β` .....(ii)
From (i) and (ii),
`(x - h)/(tan α) = (x + h)/tan β`
or `( x + h)/(x - h) = (tanβ)/(tan α)`
App. componendo and dividendo,
`( x + h + x - h )/(x + h - x + h) = (tanβ + tan α)/(tanβ - tan α)`
`( 2x)/(2h) = (tanβ + tan α)/(tanβ - tan α)`
`x = (h(tanβ + tan α))/(tanβ - tan α)`
∴ Height of the cloud is `x = (h(tanβ + tan α))/(tanβ - tan α)` ....Hence proved.
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