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