Question
If $$P\left( {3,\,2,\, - 4} \right),\,Q\left( {5,\,4,\, - 6} \right)$$ and $$R\left( {9,\,8,\, - 10} \right)$$ 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 :
$$3 : 2$$ externally
Solution :
Suppose $$R$$ divides $$PQ$$ in the ration $$\lambda :1.$$
Then, the coordinates of $$R$$ are
$$\left( {\frac{{5\lambda + 3}}{{\lambda + 1}},\,\frac{{4\lambda + 2}}{{\lambda + 1}},\,\frac{{ - 6\lambda - 4}}{{\lambda + 1}}} \right)$$
But, the coordinates of $$R$$ are given as $$\left( {9,\,8,\, - 10} \right)$$
$$\eqalign{
& \therefore \,\frac{{5\lambda + 3}}{{\lambda + 1}} = 9,\,\frac{{4\lambda + 2}}{{\lambda + 1}} = 8{\text{ and }} \cr
& \frac{{ - 6\lambda - 4}}{{\lambda + 1}} = - 10 \Rightarrow \lambda = - \frac{3}{2} \cr} $$
Hence, $$R$$ divides $$PQ$$ externally in the ratio $$3 : 2$$