Question

Let $A(2\hat{i}+3\hat{j}+5\hat{k}),$ $B(-\hat{i}+3\hat{j}+2\hat{k})$ and $C(\lambda \hat{i}+5\hat{j}+\mu \hat{k})$ be vertices of a triangle and the median through $A$ is equally inclined to the positive direction of coordinate axes. The value of $\lambda +3\mu$ is

Answer 1

Let $D$ be the median through $A.$
Position vector of $D$ is $\left(\frac{\lambda -1}{2}\right)\hat{i}+\left(\frac{5+3}{2}\right)\hat{j}+\left(\frac{\mu +2}{2}\right)\hat{k}$
Direction ratios of $AD$ are
$(\frac{\lambda -1}{2}-2,4-3,\frac{\mu +2}{2}-5)$
i.e., $(\frac{\lambda -5}{2},1,\frac{\mu -8}{2})$
But direction cosines of $AD$ are $(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$
$\Rightarrow \frac{\lambda -5}{2}=1=\frac{\mu -8}{2}$
$\Rightarrow \lambda =7$ and $\mu =10$