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.........

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- May 3rd 2009, 03:04 AMadhyetacollinear points
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......... - May 3rd 2009, 03:40 AMNonCommAlg
- May 3rd 2009, 06:14 AMadhyetaboth
they should be distinct as well as coillinear!

- May 3rd 2009, 06:15 AMadhyeta
i saw that now. it was basically a typo. i meant "...distinct collinear points..."

(Itwasntme) - May 3rd 2009, 06:29 AMJester
- May 3rd 2009, 06:35 AMadhyeta
- May 3rd 2009, 06:51 AMJester
- May 3rd 2009, 06:52 AMadhyeta
- May 3rd 2009, 07:08 AMJester
- May 3rd 2009, 07:11 AMadhyeta
- May 3rd 2009, 07:21 AMJester
- May 3rd 2009, 07:25 AMHallsofIvy
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. - May 3rd 2009, 07:26 AMadhyeta
distinct AND collinear

- May 3rd 2009, 07:46 AMadhyeta
- May 3rd 2009, 07:50 AMPlato
The points are distinct and collinear for $\displaystyle a=3:$$\displaystyle \;\;(3,1),~(1,3),~(2,2)$.