A vertical tower Stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of Elevation of the bottom and the top of the flag staff are `alpha and beta` respectively Prove that the height of the tower is `(htanalpha)/(tanbeta - tanalpha)`

Explanation

Let the height of the tower be H and OR=x
Given that, height of flag staff = h = FP and `anglePRO=alpha, angleFRO=beta`
Now, in `DeltaPRO`, `tanalpha = (PO)/(RO) = H/x`
`rArr x=H/(tanalpha)`
and in `DeltaFRO, tan beta= (FO)/(RO)=(FP+PO)/(RO)`
`tanbeta = (h+H)/(tanbeta)`...........(i)
From Eqs. (i) and (ii),

`H/(tanalpha) = (h+H)/(tanbeta)`
`Htanbeta = htanalpha + Htanalpha`
`rArr Htanbeta-Htanalpha = htanalpha`
`rArr H(tanbeta-tanalpha) = htanalpha rArr H=(htanalpha)/(tanbeta-tanalpha)`
Hence the required height of tower is `(htanalpha)/(tanbeta-tanalpha)` Hence Proved.