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Question: The top of a hill observed from the top and bottom of a building of height h is at the angle of elev...

The top of a hill observed from the top and bottom of a building of height h is at the angle of elevation p and q respectively. The height of the hill is

A

hcot⁡qcot⁡q−cot⁡p\frac { h \cot q } { \cot q - \cot p }

B

hcot⁡pcot⁡p−cot⁡q\frac { h \cot p } { \cot p - \cot q }

C

htan⁡ptan⁡p−tan⁡q\frac { h \tan p } { \tan p - \tan q }

D

None of these

Answer

hcot⁡pcot⁡p−cot⁡q\frac { h \cot p } { \cot p - \cot q }

Explanation

Solution

Let AD be the building of height h and BP be the hill, then

tan⁡q=h+xy\tan q = \frac { h + x } { y } and tan⁡p=xy\tan p = \frac { x } { y }

⇒ xcot⁡p=(h+x)cot⁡qx \cot p = ( h + x ) \cot q

⇒ h+x=hcot⁡pcot⁡p−cot⁡qh + x = \frac { h \cot p } { \cot p - \cot q }