question_answer

ABC is a triangle in a plane with vertices A(2,3,5), B(-1,3,2) and C\[(\lambda ,\,\,5,\,\,\mu )\]. If the median through A is equally inclined to the coordinate axes, then the value of (\[({{\lambda }^{3}}+{{\mu }^{3}}+5)\]is   JEE Main Online Paper (Held On 10 April 2016)

A) 676                                           

B) 1130

C) 1348         

D) 1077

Correct Answer: C

Solution :

                \[D=\left( \frac{-1+\lambda }{2},4,\frac{2+\mu }{2} \right)\] direction cosine of \[AD=\left\{ \frac{-1+\lambda }{2}-2,4,\frac{2+\mu }{2}-5 \right\}\] \[\left\{ \frac{-1+\lambda }{2}-2,4-3,\frac{2+\mu }{2}-5 \right\}\] \[\overrightarrow{AD}=\frac{\lambda -5}{2}i+j+\frac{\mu -8}{2}\widehat{k}\] \[\Rightarrow \,\,\frac{\left( \frac{\lambda -5}{2} \right)}{\sqrt{{{\left( \frac{\lambda -5}{2} \right)}^{2}}+{{1}^{2}}+{{\left( \frac{\mu -8}{2} \right)}^{2}}}}=\frac{1}{{{\sqrt{{{\left( \frac{\lambda -5}{2} \right)}^{2}}+1+\left( \frac{\mu -8}{2} \right)}}^{2}}}\] \[=\frac{\left( \frac{\mu -8}{2} \right)}{\sqrt{{{\left( \frac{\lambda -5}{2} \right)}^{2}}+1+{{\left( \frac{\mu -8}{2} \right)}^{2}}}}\] \[\overrightarrow{AD}.\,i\,=\,\overrightarrow{AD}.\,\,\widehat{j}=\overrightarrow{AD}\,.\,\widehat{k}\] \[\lambda =7,\,\,\mu =10\] \[{{\lambda }^{^{3}}}+{{\mu }^{3}}+5=343+1000+5=1348\]