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  4. If P(3, 2, - 4), Q(5, 4, - 6) and R(9, 8, - 10) are collinear, then R divides PQ in the ratio

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Q. If $P(3, 2, - 4)$, $Q(5, 4, - 6)$ and $R(9, 8, - 10)$ are collinear, then $R$ divides $PQ$ in the ratio

Introduction to Three Dimensional Geometry

Solution:

Let $R(9, 8 , -10)$ divides the join of $P(3, 2, -4)$ and $Q(5, 4, - 6)$ in the ratio $k : 1$.
Then, $9=\frac{5k+3}{k+1}$
$\Rightarrow 9k+9=5k+3$
$\Rightarrow 4k=-6$
$\Rightarrow k = \frac{-3}{2}$
$\therefore $ Required ratio is $3 : 2$ externally.