Question

Determine the values of a and b so that the points (a,b,3),(2,0,-1) and (1,-1,-3) are collinear.

Answer 1

If the three points A, B and C are collinear, then the direction ration of AB is proportional to that of BC is proportional to that of CA
A(a,b,3)
B(2,0,-1)
C(1,-1,3)
Direction ratio of AB =(2-a,-b,-1-3) =(2-a,-b,-4)
Direction ratio of BC =(1-2,-1,3+1) = (-1,-1,4)
Direction ratio of AC =(1-a,-1-b,3-3) = (1-a,-1-b,0)

Now we have
2-a/-1=-b/-1=-4/4
-2+a=-1
a=-1+2
a=1
b=-1
We would get the same answer for a and b on comparing direction ratios of AC and DC

So value of
a=1,b=-1


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