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

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

AMUAMU 2016Introduction to Three Dimensional Geometry

Solution:

Let $Q$ divides $PR$ externally in $K : 1$
image
By external section formula, we get
$ 5 = \frac{9 \times K - 1\times 3}{K-1}$
$\Rightarrow 5K - 5 = 9K - 3$
$\Rightarrow 5K - 9K = - 3 + 5$
$\Rightarrow - 4K = 2$
$\Rightarrow K = - \frac{2}{4}$
$\Rightarrow K = - \frac{1}{2}$
Hence, $Q$ divides $PR$ in the ratio $1 : 2$ (externally).