A tree standing on a horizontal plain is leaning towards east .At two points situated at distance a and b exactly due west of it.,the angle of elevation of top of it are alpha and beta respectively. prove that the height of top of tree from ground is
(b- a) tan alpha tan beta /tan alpha-tan beta
Let PQ be the leaning tree and R and S be the two given points at distance a and b from P respectively.
Let PT = x and QT = h
In ∆PQT
In ∆QRT,
Putting the value of x from (1) in (2), we get
⇒ tan α (h + a tan θ) = h tan θ
h tan α + a tan θ tan α = h tan θ
⇒ h tan α = h tan θ – a tan θ tan α
h tan α = tan θ (h – a tan α)
Now, In ∆QST
Substituting the value of tan θ from (3), we get
⇒ h2 tan β – ah tan α tan β + bh tan α tan β – h2 tan α = 0
⇒ h (h tan β – h tan α + b tan α tan β – a tan α tan β) = 0
⇒ h (tan β – tan α) + tan α tan β (b – a) = 0