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  4. The points (5, 2, 4), (6, -1, 2) and (8,-7, k) are collinear if k is equal to

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Q. The points $(5, 2, 4), (6, -1, 2) $ and $(8,-7, k)$ are collinear if $k$ is equal to

Introduction to Three Dimensional Geometry

Solution:

Let $ A (5, 2, 4), B (6, - 1, 2), C (8, -7, k)$ be the given points.
Direction Ratios of $AB$ are $< 6 - 5, -1 - 2, 2 - 4 > \, i.e., < 1 , -3 , -2 >$
Direction Ration of BC are $< 8, -6 , -7 , + 1\, k - 2 >$
i.e., $< 2 , - 6 , k - 2 >$
Since $A,B , C$ are Collinear $\therefore $ $\frac{2}{1} = \frac{-6 }{-3} = \frac{k -2}{-2}$
$\therefore $ $k -2 = - 4 \, \Rightarrow \, k = 2 - 4 = -2$.