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When observed from top of a tower, the angle of depression of two houses $A$ and $B$ in the eastern and western directions is 30 $°$ and $60 °$ respectively, then

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House

A

is nearer to the tower than house

B

House

B

is nearer to the tower than house

A

House

A

and

B

are equidistant from the tower

None of the above

answer is B.

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Detailed Solution

It is given that the angle of depression when one observes of two houses

A

and

B

in the eastern and western directions from top of a tower are 30

°

and

60 °

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

Let

MN

be the tower of height

h

.
Houses

A

and

B

are at distances

x

and

y

from the tower respectively.
In

Δ A M N,

tan θ = P e r p e n d i c u l a r b a s e

\tan 30 ° = \frac{MN}{AN}

\Rightarrow \frac{1}{\sqrt{3}} = \frac{h}{x}

∵ tan 30 ° = 13

\Rightarrow \frac{x}{\sqrt{3}} = h . . . (1)

Δ B M N,

tan θ = P e r p e n d i c u l a r b a s e

\tan 60 ° = \frac{MN}{BN}

\Rightarrow \sqrt{3} = \frac{h}{y}

∵ tan 60 ° = 3

\Rightarrow \sqrt{3} y = h . . . . (2)

From (1) and (2), we write

\frac{x}{\sqrt{3}} = \sqrt{3} y

⟹ x = 3 y

Thus, distance of

A

from tower is

3

times the distance of

B

from the tower.
Hence, house

B

is nearer to tower than house

A

.
So the correct option is 2.

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When observed from top of a tower, the angle of depression of two houses A and B in the eastern and western directions is 30° and 60° respectively, then

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When observed from top of a tower, the angle of depression of two houses $A$ and $B$ in the eastern and western directions is 30 $°$ and $60 °$ respectively, then