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).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).