Question

ABC is a triangle where $A=(2,3,5)$, $B=(-1,3,2)$ and $C(\lambda ,5,\mu )$, if the median through A is equal inclined to the positive axes, then ${\lambda }^3+{\mu }^3+5$ is?

• A. $1348$
• B. $1346$
• C. $1315$
• D. $1349$

Answer 1

The correct option is A $1348$

$A(2,3,5),B(-1,3,2),C(\lambda ,5,\mu )\therefore M=(\frac{\lambda -1}{2},4,\frac{\mu +2}{2})\therefore \overrightarrow{M}=\frac{(\lambda -1)}{2}\hat{i}+4\hat{j}+\left(\frac{\mu +2}{2}\right)\hat{k}\overrightarrow{AM}=\frac{(\lambda -5)}{2}\hat{i}+\hat{j}+\left(\frac{\mu -8}{2}\right)\hat{k}$

Since,$\overrightarrow{AM}$ equally inclined to axes.

$\therefore \frac{\lambda -5}{2}=1\Rightarrow \lambda =7\frac{\mu -8}{2}=1\Rightarrow \mu =10\therefore {\lambda }^3+{\mu }^3+5=1348$