Question

The angle of elevation of a cliff from a point $$A$$ on the ground and from the point $$B$$ $$100\ m$$ vertically above $$A$$ are $$\alpha$$ and $$\beta$$ respectively. The height of the cliff (in metres) is

A
100tanβcotβcotα
B
100cotβcotβcotα
C
100tanβcotα+cotβ
D
100cotβcotα+cotβ

Solution

The correct option is B $$\displaystyle \frac{100\cot\beta}{\cot\beta-\cot\alpha}$$$$\tan \alpha = \dfrac{h}{d}$$$$\tan \beta = \dfrac{h-100}{d}$$$$\Rightarrow \dfrac{\tan \alpha}{\tan \beta } = \dfrac{h}{h-100}$$$$\Rightarrow \dfrac{\tan \alpha}{\tan \beta } = \dfrac{h}{h-100}$$$$\Rightarrow \dfrac{\tan \beta}{\tan \alpha} = 1-\dfrac{100}{h}$$$$\Rightarrow \dfrac{\tan \alpha - \tan \beta}{\tan \alpha} = \dfrac{100}{h}$$$$\Rightarrow h = \dfrac{100\tan \alpha}{\tan \alpha-\tan \beta}$$or$$h = \dfrac{100\cot \beta}{\cot \beta-\cot \alpha}$$Mathematics

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