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 is (in metre)

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

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

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

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

Correct Answer: C

Solution :

[c] Suppose the height of the building = BC Then, in \[\Delta ABC\] \[\cot y=\frac{AB}{BC}\]\[\Rightarrow \]\[AB=BC\cdot cot\,y\]       (i) and in \[EDC,\]\[\cot x=\frac{ED}{CD}\] From Eq. (i), we get \[\cot x=\frac{BC\cot y}{CD}\]               \[(\because \,\,ED=AB)\] \[\Rightarrow \]   \[CD\cot x=h\cot y+CD\cot y\] \[(\because BC=h+CD)\] \[\Rightarrow \]   \[CD=\frac{h\cot y}{\cot x-\cot y}\] \[\because \]       \[BC=h+CD=\frac{h\cot y}{\cot x-\cot y}\] \[\Rightarrow \]   \[BC=\frac{h\cot x}{\cot x-\cot y}\].