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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

ConceptDirection Cosines and Direction Ratios of a Line

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

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

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

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

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

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

$\lambda = - \frac{3}{2}$

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

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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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