A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height $h$ . At a point on the plane, the angle of elevation of the bottom of the flagstaff is $α$ and that of the top of the flagstaff is $β$ Then the
height of the tower is :

\frac{h}{tan α}

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\frac{h tan α}{tan β - tan α}

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\frac{tan β}{h}

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\frac{tan α}{tan β - tan α}

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Solution

The correct option is D $\frac{h tan α}{tan β - tan α}$
Let $B C$ be the tower and $C D$ be the flagstaff.
The angle of elevation of the bottom of the flagstaff is $α$ and that of the top of flagstaff is $β$ .
Let $h$ be the height of the tower
In $△ A B C$ , we have
$\Rightarrow tan α = \frac{B C}{A B}$ ....{i}
In $△ A B D$
$\Rightarrow tan β = \frac{B D}{A B}$
$\Rightarrow \frac{B C + h}{A B} = \frac{B D}{A B}$ ....{ii}
Now dividing {ii} by {i}, we get
$\frac{B C + h}{B C} = \frac{tan β}{tan α}$
$\Rightarrow (B C + h) tan α = B C tan β$
$\Rightarrow B C (tan β - tan α) = h tan α$
$\Rightarrow B C = \frac{h tan α}{tan β - tan α}$

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