question_answer

If the angles of elevation of the top of a tower from three collinear points A,B and C, on a line leading to the foot of the tower, are 300, 450 and 600 respectively, then the ratio, AB : BC, is : [JEE Main Solved Paper-2015 ]

A) \[1:\sqrt{3}\]                    

B) \[2:3\]

C) \[\sqrt{3}:1\]                    

D) \[\sqrt{3}:\sqrt{2}\]

Correct Answer: C

Solution :

from figure \[\tan {{60}^{o}}=\frac{y}{x}\Rightarrow y=\sqrt{3}x\]?(i) \[\tan {{45}^{o}}=\frac{y}{x+BC}\] \[\Rightarrow \]\[x+BC=y=\sqrt{3}x\] \[\Rightarrow \]\[BC=\left( \sqrt{3}-1 \right)x\]                                ?(ii) \[\tan {{30}^{o}}=\frac{y}{x+CA}\] \[\Rightarrow \]\[x+CA=\sqrt{3}y=3x\]\[\Rightarrow \]\[CA=2x\] \[\Rightarrow \] ?..(iii) Hence \[AB:BC=\sqrt{3}\left( \sqrt{3}-1 \right)x:\left( \sqrt{3}-1 \right)x=\sqrt{3}:1\]