$\text{ If }A(3,2,-4),B(5,4,-6)\text{ and }C(9,8,-10)\text{ are three }$$\text{ collinear point then point }C\text{ divides }AB$

$\text{ Let }C\text{ divide }AB\text{ in the ratio }\lambda :1\text{. Then }$

$C\equiv \left(\frac{5\lambda +3}{\lambda +1},\frac{4\lambda +2}{\lambda +1},\frac{-6\lambda -4}{\lambda +1}\right)=(9,8-10)$

Comparing the coordinates, we get 

$\begin{array}{l}5\lambda +3=9\lambda +9\text{ or }4\lambda =-6\\ \therefore \lambda =-3/2\end{array}$

$\text{ Also from }4\lambda +2=8\lambda +8\text{ and }-6\lambda -4=-10\lambda -10\text{, we get }$

$\text{ the same value of }\lambda \text{. }$