Q. 34.3( 3 Votes )

Show that the three points A(2, 3, 4), B(-1, 2, -3) and C(-4, 1, -10) are collinear and find the ratio in which C divides AB.

Answer :

Given: A(2, 3, 4), B(-1, 2, -3) and C(-4, 1, -10)


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(2, 3, 4), B(-1, 2, -3) and C(-4, 1, -10)


Coordinates of C using section formula:




On comparing:



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


-k + 2 = -4k – 4


4k – k = - 2 – 4


3k = -6


k = -2



2k + 3 = k + 1


2k – k = 1 – 3


k = – 2



-3k + 4 = -10(k + 1)


-3k + 4 = -10k – 10


-3k + 10k = -10 – 4


7k = -14


k = -2


The value of k is the same in all three times


Hence, A, B and C are collinear


As k = -2


C divides AB externally in ratio 2:1


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