question_answer

A tower of height b subtends an angle at a point O on the level of the foot of the tower and at a distance a from the foot of the tower. If a pole mounted on the tower also subtends an equal angle at O, the height of the pole is  [MP PET 1993, 2004]

A) \[b\,\left( \frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}} \right)\]

B) \[b\,\left( \frac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}-{{b}^{2}}} \right)\]

C) \[a\,\left( \frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}} \right)\]

D) \[a\,\left( \frac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}-{{b}^{2}}} \right)\]

Correct Answer: B

Solution :

\[\tan \alpha =\frac{b}{a}\], \[\tan 2\alpha =\frac{2(b/a)}{1-{{(b/a)}^{2}}}=\frac{p+b}{a}\] \[\Rightarrow \] \[\frac{2ba}{{{a}^{2}}-{{b}^{2}}}=\frac{p+b}{a}\Rightarrow \frac{2b{{a}^{2}}-{{a}^{2}}b+{{b}^{3}}}{{{a}^{2}}-{{b}^{2}}}=p\] \[\Rightarrow \] \[p=\frac{b({{a}^{2}}+{{b}^{2}})}{({{a}^{2}}-{{b}^{2}})}\].