The angle of elevation of the top of a tower EF (F being the foot of the tower) as seen from a point A which is on the same level as F, is `alpha`. On advancing towards the foot of the tower the angle of elevation of the top of the tower as seen from a point B such that AB=x , is `beta`. If BF=y, h is the height of the tower and `alpha + beta = (pi)/(2)`, then which one of the following is correct?

Options

A. `h^(2)=x^(2)+xy`
B. `h=y^(2)+xy^(2)`
C. `h^(2)=y^(2)+xy`
D. `h=y+x^(2)y`

Answer

C

Explanation

Let EF be the height of the tower, and FB = Y , AB = x and EF = h.
In `DeltaBEF`,
`tan beta = (EF)/(BF)`
`tan beta = (h)/(y) `...(i)

and in `Delta AFE`,
`tan alpha = (EF)/(AF)`
`rArr tan alpha = (h) /(a+y)`,
` tan ((pi)/(2)-beta)=(h)/(x+y)` (Given that `alpha + beta = (pi)/(2))`
`rArr cot beta = (h) /(x+y)` ...(ii)
From Eqs. (i) and (ii), we get
`tan beta. cot beta = (h)/(y).(h)/((x+y)) " " (.:. tan beta * cot beta=1)`
`rArr xy+y^(2)=h^(2)`