If P (2, 3, 4), Q (5, 8, 7) and R (-1, -2, 1) are collinear, then R divides PQ in the ratio:

Correct option is C

​$$\begin{aligned}&\text{Given:} \\&\quad P(2, 3, 4), \quad Q(5, 8, 7), \quad R(-1, -2, 1) \text{ are collinear.} \\[10pt]&\text{Assume that point } R \text{ divides the line segment } \overline{PQ} \text{ in the ratio } k : 1. \\[10pt]&\text{Using the section formula for the x-coordinate:} \\&\quad x = \frac{kx_2 + x_1}{k + 1} \\[10pt]&\text{Substitute values:} \\&\quad -1 = \frac{5k + 2}{k + 1} \Rightarrow -k - 1 = 5k + 2 \Rightarrow -k - 1 - 5k - 2 = 0 \Rightarrow -6k = 3 \Rightarrow k = -\frac{1}{2} \\[10pt]&\text{So, the ratio is:} \\&\quad k : 1 = -1 : 2 \\[10pt]&\text{Since a ratio cannot be negative, this means point } R \text{ divides } \overline{PQ} \text{ externally in the ratio:} \\&\quad {1 : 2}\end{aligned}$$​​