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

$(\frac{-\lambda +2}{\lambda +1},\frac{2\lambda +3}{\lambda +1}\frac{-3\lambda +4}{\lambda +1})$

But the coordinates of C are (-4, 1, -10)

$\therefore \frac{-\lambda +2}{\lambda +1}=-4,\frac{2\lambda +3}{\lambda +1}=1,\frac{-3\lambda +4}{\lambda +1}=-10$

$\Rightarrow -\lambda +2=-4\lambda -4,2\lambda +3=\lambda +1,$

$-3\lambda +4=-10\lambda -10$

$\Rightarrow 3\lambda =-6,\lambda =-2,7\lambda =-14$

$\therefore \lambda =-2,\lambda =-2,\lambda =-2$

From these three equations, we have :

$\lambda =-2$

So, C divides AB in the ratio 2 : 1 (externally).