Question

Show that the three points A(2, 3, 4), B(–1, 2 – 3) and C(–4, 1, –10) are collinear and find the ratio in which C divides AB.

Answer 1

Suppose C divides AB in the ratio $\lambda :1$.
Then, the coordinates of C are $\left(\frac{-\lambda +2}{\lambda +1},\frac{2\lambda +3}{\lambda +1},\frac{-3\lambda +4}{\lambda +1}\right)$.
But, the coordinates of C are ($-$4, 1, $-$10).
$$\begin{gathered} \therefore \frac{-\lambda +2}{\lambda +1}=-4,\frac{2\lambda +3}{\lambda +1}=1,\frac{-3\lambda +4}{\lambda +1}=-10 \\ \Rightarrow -\lambda +2=-4\lambda -4,2\lambda +3=\lambda +1,-3\lambda +4=-10\lambda -10 \\ \Rightarrow 3\lambda =-6,\lambda =-2,7\lambda =-14 \\ \therefore \lambda =-2,\lambda =-2,\lambda =-2 \end{gathered}$$

From these three equations, we have:
$\lambda =-2$
So, C divides AB in the ratio 2:1 (externally).