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A flag-staff of height h stand on the top of a school building. If the angles of elevation of the and bottom of the flag-staff have measure $α$ and $β$ are respectively from a point on the ground, then the height of the building is.

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\frac{htan β}{\tan \tan α + \tan β}

\frac{hcot β}{\tan α - \tan β}

\frac{htan α}{\tan α - \tan β}

\frac{htan β}{\tan α - \tan β}

answer is D.

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Detailed Solution

Suppose,
Height of building =AB
Height of flag staff,

CA = h

Total height,

CB = CA + AB

CB = h + AB

In the above figure, D is observer point.

In △ ABD

\tan β = \frac{AB}{BD}

BD = ABcot β \dots \dots \dots \dots \dots (1)

Similarly,

in Δ CBD

\tan α = \frac{CB}{BD}

BD = CBcot α

BD = (h + aB) \cot α \dots \dots \dots (2)

From equation 1 and 2,

ABcot β = (h + AB) \cot α

ABcot β = hcot α + ABcot α

AB (\cot β - \cot α) = hcot α

AB = \frac{hcot α}{\cot β - \cos α}

AB = \frac{\frac{h}{\tan α}}{\frac{\tan α - \tan β}{\tan β \cdot \tan α}}

AB = h (\frac{\tan β}{\tan α - \tan β})

Correct option is 4.

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