Question

Using section formula, prove that the three points A(-2,3,5), B(1,2,3) and C(7,0,-1) are collinear.

Answer 1

Suppose the given points are collinear and C divides AB in the ratio $\lambda :1$. Then, coordinates of C are

$\frac{\lambda -2}{\lambda +1},\frac{2\lambda +3}{\lambda +1},\frac{3\lambda +5}{\lambda +1}$

But, coordinates of C are (7,0,-1).

So, $\frac{\lambda -2}{\lambda +1}=7,\frac{2\lambda +3}{\lambda +1}=0$ and $\frac{3\lambda +5}{\lambda +1}=-1$

From each of these equations, we get: $\lambda =-\frac{3}{2}$.

Since each of these equations give the same value of $\lambda$. Therefore, the given points are collinear and C divides AB externally in the ratio 3:2.