If the points A(3, 2, -4), B(9, 8, -10), and C(5, 4, -6) are collinear, what ratio does point C divide AB in?
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Question

If the points A(3, 2, -4), B(9, 8, -10), and C(5, 4, -6) are collinear, what ratio does point C divide AB in?

Accepted Answer

Let C divide AB in the ratio $k:1$. Using the section formula for x-coordinate: $5 = \frac{k(9) + 1(3)}{k+1} \implies 5k + 5 = 9k + 3 \implies 4k = 2 \implies k = 1/2$. Thus, the ratio is $1:2$ internally.

Introduction to Three Dimensional Geometry

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