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Question
If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio
Options
3 : 2 externally
3 : 2 internally
2 : 1 internally
2 : 1 externally
Solution
3: 2 externally
\[\text{ Suppose the point R divides PQ in the ratio } \lambda: 1 . \]
\[\text{ Coordinates of R are } \left( \frac{5\lambda + 3}{\lambda + 1}, \frac{4\lambda + 2}{\lambda + 1}, \frac{- 6\lambda - 4}{\lambda + 1} \right) . \]
\[\text { But the coordinates of R are } \left( 9, 8, - 10 \right) . \]
\[ \therefore \frac{5\lambda + 3}{\lambda + 1} = 9, \frac{4\lambda + 2}{\lambda + 1} = 8 \text{ and } \frac{- 6\lambda - 4}{\lambda + 1} = - 10\]
\[\text{ From each of these equations, we get }\]
\[\lambda = - \frac{3}{2}\]
\[ \therefore \text{ R divides PQ in the ratio 3: 2 externally } .\]
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