• # question_answer A balloon is observed simultaneously from three points A, B and C on a straight road directly under it. The angular elevation at B is twice and at C is thrice that of A. If the distance between A and B is 200 metres and the distance between B and C is 100 metres, then the height of balloon is given by [Roorkee 1989] A) 50 metres B) $50\,\sqrt{3}$ metres C) $50\,\sqrt{2}$ metres D) None of these

$x=h\cot 3\alpha$    .....(i) $(x+100)=h\cot 2\alpha$ ......(ii) $(x+300)=h\cot \alpha$ ......(iii) From (i) and (ii), $-100=h\,(\cot 3\alpha -\cot 2\alpha ),$ From (ii) and (iii), $-200=h(\cot 2\alpha -\cot \alpha ),$ $100=h\,\left( \frac{\sin \alpha }{\sin 3\alpha \sin 2\alpha } \right)$ and $200=h\,\left( \frac{\sin \alpha }{\sin 2\alpha \sin \alpha } \right)$ or $\frac{\sin 3\alpha }{\sin \alpha }=\frac{200}{100}\Rightarrow \frac{\sin 3\alpha }{\sin \alpha }=2$ $\Rightarrow$ $3\sin \alpha -4{{\sin }^{3}}\alpha -2\sin \alpha =0$ $\Rightarrow$ $4{{\sin }^{3}}\alpha -\sin \alpha =0\Rightarrow \sin \alpha =0$ or ${{\sin }^{2}}\alpha =\frac{1}{4}={{\sin }^{2}}\left( \frac{\pi }{6} \right)\Rightarrow \alpha =\frac{\pi }{6}$ Hence, $h=200\sin \frac{\pi }{3}=200\frac{\sqrt{3}}{2}=100\sqrt{3}$,  {from (ii)} .