Question

ABC is a triangle in a plane with vertices A(2, 3, 5), B( -1, 3, 2) and $C(\lambda ,5,\alpha ).$ If the median through A is equally inclined to the coordinate axes, then the value of ${(\lambda }^3+{\mu }^3+5)$ is :

• A. 1130
• B. 1348
• C. 676
• D. 1077

Answer 1

The correct option is B 1348

$M=\frac{-1+\lambda }{2},\frac{3+5}{2},\frac{2+2}{2}$

$\Rightarrow M\equiv \left(\frac{\lambda -1}{2},4,\frac{\alpha +2}{2}\right)$

$DirectionratioofAM$

$=\left(\frac{\lambda -1}{2}-2,4-3,\frac{\lambda +2}{2}-5\right)$

$=\left(\frac{\lambda -5}{2},1,\frac{\lambda -8}{2}\right)$

$medianAMisequalyinclinetoco-oridnateaxies.$

$\therefore Directionratio=k\left(1,1,1\right)$

$so,$

$k=1,\frac{\lambda -5}{2}=1,\frac{\alpha -8}{2}=10$

$\Rightarrow \lambda =7,\alpha =10$

$Here,$

$\left({\lambda }^3+{\alpha }^3+5\right)$

$=7^3+{10}^3+5=1348$

$Hence,theoptionBisthecorrectanswer.$