Question

$ABC$ 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:

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

Answer 1

The correct option is A $1348$


Coordinates of $D=(\frac{\lambda -1}{2},4,\frac{\mu +2}{2})$
$\therefore \overrightarrow{AD}=(\frac{\lambda -1}{2}-2)\hat{i}+(4-3)\hat{j}+(\frac{\mu +2}{2}-5)\hat{k}$
$=\left(\frac{\lambda -5}{2}\right)\hat{i}+\hat{j}+\left(\frac{\mu -8}{2}\right)\hat{k}$
Since, $\overrightarrow{AD}$ makes equal angle with the coordinate axes, the direction ratios are equal.
$\Rightarrow$ Direction ratios are equal.
$\text{i.e.,}\frac{\lambda -5}{2}=1=\frac{\mu -8}{2}$
$\Rightarrow \lambda =7,\mu =10$
$\therefore {\lambda }^3+{\mu }^3+5=1348$