Prove that the points A(2,0,3),B(3,2,-1) and C(1,-2,-5) are collinear. Sir, please note that I have solved this question, but collinearity of the points is not proved. The question seems to be wrong. So please solve and determine whether the question is right or wrong.

To show collinearity between A,B and C, lets assume that B divides AC in the ratio of k:1

 

Hence, using the section formula

So, for x coordinates

3 =( 1*k+2)/ (k+1)

3k+3 = k+2

2k = -1

k = -1/2

 

Validating the y coordinate of B using same

y = (-2(k) + 0)/(k+1)

y =  (-2(-1/2) + 0)/(-1/2+1)

y = 1/(1/2) = 2 which is correct

 

Similarly, validating the z coordinate of B 

z =  (-5(k) + 3)/(k+1)

z = (-5(-1/2) + 3)/(-1/2+1)

z = (11/2)(1/2) = 11

which is wrong as z coordinate should be equal to -1. 

 

Hence, the question is wrong. The three points are not collinear.