Question

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

Answer 1

Suppose C divides AB in the ratio \[\lambda: 1\]

Then, coordinates of C are as follows: 

\[\left( \frac{9\lambda + 3}{\lambda + 1}, \frac{8\lambda + 2}{\lambda + 1}, \frac{- 10\lambda - 4}{\lambda + 1} \right)\]

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

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