Question

If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio
(a) 3 : 2 internally
(b) 3 : 2 externally
(c) 2 : 1 internally
(d) 2 : 1 externally

Answer 1

$$\begin{gathered} \left(\mathrm{b}\right)3:2\mathrm{externally} \\ \mathrm{Suppose}\mathrm{the}\mathrm{point}R\mathrm{divides}PQ\mathrm{in}\mathrm{the}\mathrm{ratio}\lambda :1. \\ \mathrm{Coordinates}\mathrm{of}R\mathrm{are}\left(\frac{5\lambda +3}{\lambda +1},\frac{4\lambda +2}{\lambda +1},\frac{-6\lambda -4}{\lambda +1}\right). \\ \mathrm{But}\mathrm{the}\mathrm{coordinates}\mathrm{of}R\mathrm{are}\left(9,8,-10\right). \\ \therefore \frac{5\lambda +3}{\lambda +1}=9,\frac{4\lambda +2}{\lambda +1}=8\mathrm{and}\frac{-6\lambda -4}{\lambda +1}=-10 \\ \mathrm{From}\mathrm{each}\mathrm{of}\mathrm{these}\mathrm{equation}s,\mathrm{we}\mathrm{get} \\ \lambda =-\frac{3}{2} \\ \therefore R\mathrm{divides}PQ\mathrm{in}\mathrm{the}\mathrm{ratio}3:2\mathrm{externally}. \end{gathered}$$