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)`

Options

A. `(h tan beta)/(tan alpha-tan beta)`
B. `(h tan beta)/(tan alpha + tan beta)`
C. `(h cos beta)/(cos alpha - cos beta)`
D. `(h)/(cos (alpha-beta))`

Answer

A

Explanation

Let BC be the vertical tower and CD be the flagstaff so that CD=h
Let P be the point of observation on the plane.
Then, `angleBPC=beta and angleBPD=alpha`
Let BC=x
Now, `(PB)/(x)=cot beta rArr PB=x cot beta ...(1)`
and `(PB)/(x+h)=cot alpha rArr PB = (x+h) cot alpha " " ...(2)`
From (1) and (2), we get
`x cot beta = (x+h) cot alpha rArr x (Cot beta - cot alpha)=h cot alpha`
`:.` Height of the tower,
`x=(h cot alpha)/(cot beta - cot alpha)=(h tan beta)/(tan alpha-tan beta)`