question_answer

The angles of elevation of the top of a building from the top and bottom of a tree are x and y. respectively. If the height of the tree is h m, then the height of the building (in metre) is

A) \[\frac{h\cot x}{\cos x+\cot y}\]             

B) \[\frac{h\cot \,y}{\cos x+\cot y}\]

C) \[\frac{h\cot \,x}{\cot x-\cot y}\]             

D) \[\frac{h\cot \,y}{\cot x-\cot y}\]

Correct Answer: C

Solution :

Let height of tree be h m and height of building be b m.

In \[\Delta AED,\]            \[\tan x=\frac{AE}{ED}\]

\[\Rightarrow \]   \[\tan x=\frac{b-h}{ED}\]

\[\Rightarrow \]   \[ED=(b-h)\cot x\]                       (i)

From \[\Delta ABC,\]\[\tan y=\frac{AB}{BC}\]

\[\Rightarrow \]   \[\tan y=\frac{b}{BC}\]

\[\Rightarrow \]   \[BC=b\cot y\]                           (ii)

From Eqs. (i) and (ii), we get

\[BC=ED\]

\[\therefore \]      \[(b-h)\cot x=b\cot y\]

\[\Rightarrow \]   \[b\cot x-h\cot x=b\cot y\]

\[\Rightarrow \]   \[b\,(\cot x-\cot y)=h\cot x\]

\[\therefore \]                  \[b=\frac{h\cot x}{\cot x-\cot y}\]