question_answer

A vertical pole \[PS\] has two marks at \[Q\] and \[R\] such that the portions \[PQ,\,\,\,PR\] and \[PS\] subtend angles \[\alpha ,\,\,\,\beta \] and \[\gamma \] at a point on the ground distance \[x\] from the bottom of pole. If \[PQ=a,\,\,PR=b,\,\,PS=c\]and\[\alpha +\beta +\gamma ={{180}^{o}}\], then \[{{x}^{2}}\] is equal to

A) \[\frac{{{a}^{3}}}{a+b+c}\]                          

B) \[\frac{{{b}^{3}}}{a+b+c}\]

C) \[\frac{{{c}^{3}}}{a+b+c}\]                          

D) \[\frac{abc}{a+b+c}\]

Correct Answer: A

Solution :

We have\[\tan \alpha =\frac{a}{x},\,\,\tan \beta =\frac{b}{x}\]and\[\tan \gamma +\frac{c}{x}\] \[\therefore \]  \[\alpha +\beta +\gamma ={{180}^{o}}\] So,\[\tan \alpha +\tan \beta +\tan \gamma =\tan \alpha \cdot \tan \beta \cdot \tan \gamma \] or            \[\frac{a}{x}+\frac{b}{x}+\frac{c}{x}=\frac{a}{x}\cdot \frac{b}{x}\cdot \frac{c}{x}\] or            \[{{x}^{2}}=\frac{abc}{a+b+c}\]