Answer :

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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