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From the top of a lighthouse, the angles of depression of two stations on opposite sides and $a$ distance apart are $α$ and $β$ . The height of the lighthouse is

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\frac{a}{cotαcotβ}

\frac{a}{cotα + cotβ}

\frac{acotαcotβ}{cotα + cotβ}

\frac{atanαtanβ}{cotα + cotβ}

answer is B.

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Detailed Solution

It is given that from the top of a lighthouse, the angles of depression of two stations are

α

and

β

which are situated at a distance

.
According to the given data the figure has been drawn below.
Question Image

Let the height of light house be

h = AD

, as shown in the figure above and

∠ ABD = α (Given)

∠ ACD = β (Given)

Now, in

△ ABD,

\frac{BD}{AD} = cotα [cotα = \frac{Adjacent Side}{Opposite Side}]

BD = hcotα

Also, in

△ ADC,

\frac{CD}{AD} = cotβ [cotβ = \frac{Adjacent Side}{Opposite Side}]

CD = hcotβ

Given that,

= a

∴ a = BC

\Rightarrow a = BD + CD

\Rightarrow a = hcotα + hcotβ

\Rightarrow a = h (cotα + cotβ)

\Rightarrow h = \frac{a}{cotα + cotβ}

So the height of the lighthouse is

\frac{a}{cotα + cotβ}

.
Hence, option 2 is correct.

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