11. Given that P (3, 2, – 4), Q (5, 4, – 6) and R (9, 8, –10) are collinear. Find the ratio in which Q divides PR.

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Payal Gupta | Contributor-Level 10

8 months ago

11. Let point Q divides PR in the k : 1. Then co-ordinate of Q will be

$(\frac{k (9) + 1 (3)}{k + 1}, \frac{k (8) + 1 (2)}{k + 1}, \frac{k (- 10) + 1 (- 4)}{k + 1})$

=> (5, 4, –6) = $\cdot (\frac{9 k + 3}{k + 1} \cdot, \cdot \frac{8 k + 2}{k + 1} \cdot, \cdot \frac{- 10 k - 4}{k + 1} \cdot)$

Equating the co-ordinates we get,

$\frac{9 k + 3}{k + 1}$ = 5

=> $9 k + 3 = 5 (k + 1)$

=> $9 k + 3 = 5 k + 5$

=> $9 k - 5 k = 5 - 3$

=> $4 k = 2$

=> $k = \frac{2}{4}$

=> $k = \frac{1}{2}$

Putting $k = \frac{1}{2}$ in y-coordinate and z-coordinate

$\frac{8 k + 2}{k + 1}$

= $\frac{(8 x \frac{1}{2}) + 2}{\frac{1}{2} + 1}$

= (4 + 2) ÷ $\frac{3}{2}$

= 6 x $\frac{2}{3}$

= 4

And

$\frac{(- 1 0x \frac{1}{2}) - 4}{\frac{1}{2} + 1}$

= (–5 – 4) ÷ $\frac{3}{2}$

= – 9 x $\frac{2}{3}$

= – 6

Which is matching with the given co-ordinates of Q.

Hence, the rati

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