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

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Solution

Given points $P (3, 2, - 4), Q (5, 4, - 6)$ and $R (9, 8, - 10)$
Let $Q$ divides $P R$ in the ratio $k : 1$ .
So, by section formula, coordinates of $Q$ are $(\frac{k (9) + 1 (3)}{k + 1}, \frac{k (8) + 1 (2)}{k + 1}, \frac{k (- 10) + 1 (- 4)}{k + 1}$
$= (\frac{9 k + 3}{k + 1}, \frac{8 k + 2}{k + 1}, \frac{- 10 k + 4}{k + 1})$
But the given coordinates of $Q$ are $(5, 4, - 6)$ .
On comparing $\frac{9 k + 3}{k + 1} = 5$
$\Rightarrow 9 k + 3 = 5 k + 5$
$\Rightarrow k = \frac{1}{2}$
So, $Q$ divides $P Q$ in the ratio $1 : 2$ .

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