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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 - CBSE (Arts) Class 12 - Mathematics

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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

  • 3 : 2 externally

  •  3 : 2 internally

  •  2 : 1 internally

  •  2 : 1 externally

     

Solution

3: 2 externally

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

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

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

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

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

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

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

  Is there an error in this question or solution?

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Solution 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 Concept: Direction Cosines and Direction Ratios of a Line.
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