Q. 48

# A tree standing on a horizontal plane is leaning towards east. At two points situated at distances a and b exactly due west on it, the angles of elevation of the top are respectively α and β. Prove that the height of the top from the ground is

In the fig let RP be the leaning tree, R & S be the two points at distance ‘a’ and ‘b’ from point Q.

In ∆PQR

tan θ° =

tan θ° =

x = ………….(1)

In ∆PQS

tan α =

tan α = ……………….(2)

In ∆PQT

tan α =

tan β =

tan β = …………….(3)

On substituting value of x from eqn (1) in eqn (2) we get,

tan α =

h tan α + a tan θ tan α = h tan θ

h tan α = tan θ(h-a tan α)

tan θ = …………….(4)

Now on substituting value of x in eqn (3)

tan β =

Now on substituting value of tan θ in eqn (4)

tan β =

h2tan β = 0

h(h tan β = 0

h()+tan αtan β(b-a) = 0

h = Proved

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