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 |
\[BC=ED\] |
\[\therefore \] \[(b-h)\cot x=b\cot y\] |
\[\Rightarrow \] \[b\cot x-h\cot x=b\cot y\] |
\[\Rightarrow \] \[b\,(\cot x-\cot y)=h\cot x\] |
\[\therefore \] \[b=\frac{h\cot x}{\cot x-\cot y}\] |
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