Question

If the points A(3, 2, -4), B (9, 8, -10) and C(5, 4, -6) are collinear, find the ratio in which C divides AB.

Answer 1

Suppose C divides AB in the ratio \(\lambda : 1)

Then, coordinates of C are as follows :

$(\frac{9\lambda +3}{\lambda +1},\frac{8\lambda +2}{\lambda +1},\frac{-10\lambda -4}{\lambda +1})$

But, the coordinates of C are (5, 4, -6)

$\therefore \frac{9\lambda +3}{\lambda +1}=5,\frac{8\lambda +2}{\lambda +1}=4,\frac{-10\lambda -4}{\lambda +1}=-6$

These three equations give $\lambda =\frac{1}{2}$

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