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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:

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\frac{h}{tanα}

\frac{htanα}{tanβ - tanα}

\frac{tanβ}{h}

\frac{tanα}{tanβ - tanα}

answer is B.

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Detailed Solution

Given, height of flagstaff= h
The angle of elevation of the bottom of the flagstaff is

α

and the top of the flagstaff is

β .

We have to find the height of the tower .
Let

BC

be the tower and

CD

be the flagstaff.
Let

h

be the height of the tower.
Question Image

We know that tan

θ = \frac{perpendicular}{base}

.
In

△ ABC

, we have

\Rightarrow tanα = \frac{BC}{AB}

.... {i}
In

△ ABD

\Rightarrow tanβ = \frac{BD}{AB}

\Rightarrow \frac{BC + h}{AB} = \frac{BD}{AB}

.... {ii}
Now dividing {ii} by {i},

\Rightarrow \frac{BC + h}{BC} = \frac{tanβ}{tanα}

\Rightarrow (BC + h) tanα = BCtanβ

\Rightarrow BCtanα + htanα = BCtanβ

\Rightarrow htanα = BCtanβ - BCtanα

\Rightarrow BC (tanβ - tanα) = htanα

\Rightarrow BC = \frac{htanα}{tanβ - tanα}

Hence, the height of the tower is

\frac{htanα}{tanβ - tanα}

.
Therefore, the correct option is 2.

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