Question

If the points $(5,4,2),(8,k,-7)$ and $(6,2,-1)$ are collinear, then the value of $|k|=$

Answer 1

Given: $A(5,4,2)$ and $C(6,2,-1)$
Let $B(8,k,-7)$ divides the segment $AC$ in $\lambda :1$
$\Rightarrow 8=\frac{6\lambda +5}{\lambda +1}\Rightarrow \lambda =-\frac{3}{2}$
The negative ratio indicates that Point $B$ divides the segment $AC$ externally in ratio $3:2$
$\Rightarrow k=\frac{3\cdot 2-2\cdot 4}{3-2}=-2$
$\therefore |k|=2$