question_answer

If \[\alpha ,\,\,\beta \] and \[\gamma \] are the altitudes of the \[\Delta ABC\] from the vertices A, B and C respectively, then value of \[4{{\Delta }^{2}}\,\left( \frac{1}{{{\alpha }^{2}}}+\frac{1}{{{\beta }^{2}}}+\frac{1}{{{\gamma }^{2}}} \right)\] is

A)  \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]            

B)  \[\frac{1}{2}\,({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\]

C)  \[{{a}^{2}}-{{b}^{2}}+{{c}^{2}}\]              

D)  \[\frac{3}{\Delta }\]

Correct Answer: A

Solution :

 Clearly, \[{{X}_{CM}}\] \[{{m}_{2}}{{x}_{2}}=-{{m}_{1}}{{x}_{1}}\] \[\therefore \] \[{{x}_{2}}=\frac{{{m}_{1}}}{{{m}_{2}}}{{x}_{1}}\]