Q. 65.0( 1 Vote )

# If the points A(3

Given: A(3, 2, -4), B(9, 8, -10) and C(5, 4, -6)

To prove: A, B and C are collinear

To find: the ratio in which C divides AB

Formula used:

Section Formula:

A line AB is divided by C in m:n where A(x, y, z) and B(a, b, c). The coordinates of C is given by, Let C divides AB in ratio k: 1

Three points are collinear if the value of k is the same for x, y and z coordinates

Therefore, m = k and n = 1

A(3, 2, -4), B(9, 8, -10) and C(5, 4, -6) Coordinates of C using section formula:  On comparing: 9k + 3 = 5(k + 1)

9k + 3 = 5k + 5

9k – 5k = 5 – 3

4k = 2   8k + 2 = 4(k + 1)

8k + 2 = 4k + 4

8k – 4k = 4 – 2

4k = 2   -10k – 4 = -6(k + 1)

-10k – 4 = -6k – 6

-10k + 6k = 4 – 6

-4k = -2  The value of k is the same in all three times

Hence, A, B and C are collinear C divides AB externally in ratio 1:2

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