# Math Help - collinear points

1. ## collinear 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.........

2. Originally Posted by adhyeta

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.........
distinct or collinear? (why am i asking this? because the title of your post is "collinear points"!!)

3. ## both

they should be distinct as well as coillinear!

4. i saw that now. it was basically a typo. i meant "...distinct collinear points..."

5. Originally Posted by adhyeta
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.........
Clearly $a \ne 1$. Create 2 vectors connecting your points and the require that they be parallel. This will give you your $a$ value.

6. Originally Posted by danny arrigo
Clearly $a \ne 1$. Create 2 vectors connecting your points and the require that they be parallel. This will give you your $a$ value.
hello danny,

we basically need the number of values of a & not the value of a.

7. Originally Posted by adhyeta
hello danny,

we basically need the number of values of a & not the value of a.
But in finding the value(s) you can then answer your question. It is probably the most direct way.

8. Originally Posted by danny arrigo
But in finding the value(s) you can then answer your question. It is probably the most direct way.
well, i said that because the answer comes out to be d. (infinitely many)
how will you lead to that with that method ?

9. Originally Posted by adhyeta
well, i said that because the answer comes out to be d. (infinitely many)
how will you lead to that with that method ?
How did you arrive at that answer?

10. Originally Posted by danny arrigo
How did you arrive at that answer?
its given in the text. i did not.

11. Originally Posted by adhyeta
its given in the text. i did not.
I guess it comes back to NonCommAlg's question - distinct or collinear?

12. 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.

13. distinct AND collinear

14. Originally Posted by HallsofIvy
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.
even i thought that. the book however says infinitely many.

15. The points are distinct and collinear for $a=3:$ $\;\;(3,1),~(1,3),~(2,2)$.

Page 1 of 2 12 Last