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)`.

Answer 1

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.