question_answer

AB is a vertical tower. The point A is on the ground and C is the middle point of AB. The part CB subtend an angle \[\alpha \]at a point P on the ground. If \[AP=n\,AB,\]then the correct relation is  [MNR 1989; IIT 1980]

A) \[n=({{n}^{2}}+1)\tan \alpha \]

B) \[n=(2{{n}^{2}}-1)\tan \alpha \]

C) \[{{n}^{2}}=(2{{n}^{2}}+1)\tan \alpha \]

D) \[n=(2{{n}^{2}}+1)\tan \alpha \]

Correct Answer: D

Solution :

\[\tan \alpha =\frac{-\frac{AC}{AP}+\frac{AB}{AP}}{1+\frac{AC}{AP}.\frac{AB}{AP}}\]  \[\{AP=n(AB)\] \[\Rightarrow AP=2n(AC)\}\] \[\tan \alpha =\frac{-\frac{1}{2n}+\frac{1}{n}}{1+\frac{1}{2{{n}^{2}}}}\] \[\Rightarrow \]  \[\frac{n}{(2{{n}^{2}}+1)}=\tan \alpha \]\[\Rightarrow \]  \[n=(2{{n}^{2}}+1)\tan \alpha \].