The number of real numbers a such that (a,1),(1,a) & (a-1,a-1) are three distinct points is
a)0
b)1
c)at least 2 but finitely many
d)infinitely many
i tried putting a=-1,0,1/2,1,2 which lead me to the answer a) which isn't correct.........
If (1, a), (a, 1), and (a-1, a-1) are co-linear, then we must have (a-1)/(1- a)= -1 (the slope of the line from (1,a) to (a,1)) equal to (a-1-a/ a-1-1)= -1/a (the slope of the line from (a, 1) to (a-1, a-1)). -1= -1/a is satisfied only by a= 1. But if a= 1, (1, a)= (1, 1)= (a, 1) so the three points are not distinct.
There is NO value of a that will make these points distinct and collinear.