#### Question

If *P* (3, 2, −4), *Q* (5, 4, −6) and *R* (9, 8, −10) are collinear, then *R* divides *PQ* in the ratio

3 : 2 externally

3 : 2 internally

2 : 1 internally

2 : 1 externally

#### Solution

3: 2 externally

\[\text{ Suppose the point R divides PQ in the ratio } \lambda: 1 . \]

\[\text{ Coordinates of R are } \left( \frac{5\lambda + 3}{\lambda + 1}, \frac{4\lambda + 2}{\lambda + 1}, \frac{- 6\lambda - 4}{\lambda + 1} \right) . \]

\[\text { But the coordinates of R are } \left( 9, 8, - 10 \right) . \]

\[ \therefore \frac{5\lambda + 3}{\lambda + 1} = 9, \frac{4\lambda + 2}{\lambda + 1} = 8 \text{ and } \frac{- 6\lambda - 4}{\lambda + 1} = - 10\]

\[\text{ From each of these equations, we get }\]

\[\lambda = - \frac{3}{2}\]

\[ \therefore \text{ R divides PQ in the ratio 3: 2 externally } .\]