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