Question 55

Mathematics Heights and Distances Easy

The top of a hill observed from the top and bottom of a building of height \( h \) is at angles of elevation \( p \) and \( q \) respectively. The height of the hill is-

(A) \( \frac{h \cot q}{\cot q-\cot p} \)
(B) \( \frac{h \cot p}{\cot p-\cot q} \)
(C) \( \frac{h \tan p}{\tan p-\tan q} \)
(D) \( \frac{h \sec p}{\tan p-\tan q} \)
✓ Correct Answer: Option B Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

\[ \begin{aligned} \text{Let AD} &= h \text{ be the height of the building and EB be the height of the hill.} \\ \text{We have } \tan q &= \frac{h+x}{y} \\ \text{and } \tan p &= \frac{x}{y} \\ \Rightarrow y &= x \cot p \\ \text{Now, } \tan q &= \frac{h+x}{x \cot p} \\ \Rightarrow x \cot p &= (h+x) \cot q \\ \Rightarrow x &= \frac{h \cot q}{\cot p - \cot q} \\ \therefore h+x &= h + \frac{h \cot q}{\cot p - \cot q} \\ &= \frac{h \cot p}{\cot p - \cot q} \end{aligned} \]

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