Question

Given that P(3, 2, –4), Q(5, 4, –6) and R(9, 8, –10) are collinear. Find the ratio in which Q divides PR.

Answer 1

Let Q divide PR in the ratio $\lambda :1$
Thus, the coordinates of Q 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 Q 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 equation gives $\lambda =\frac{1}{2}$.
So, Q divides PR in the ratio $\frac{1}{2}:1$ or 1:2