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Math Help - collinear points

  1. #1
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    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.........
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    Quote Originally Posted by adhyeta View Post

    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"!!)
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    both

    they should be distinct as well as coillinear!
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    i saw that now. it was basically a typo. i meant "...distinct collinear points..."
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    Quote Originally Posted by adhyeta View Post
    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.
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    Quote Originally Posted by danny arrigo View Post
    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.
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    Quote Originally Posted by adhyeta View Post
    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.
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    Quote Originally Posted by danny arrigo View Post
    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 ?
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    Quote Originally Posted by adhyeta View Post
    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?
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    Quote Originally Posted by danny arrigo View Post
    How did you arrive at that answer?
    its given in the text. i did not.
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    Quote Originally Posted by adhyeta View Post
    its given in the text. i did not.
    I guess it comes back to NonCommAlg's question - distinct or collinear?
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    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.
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    distinct AND collinear
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    Quote Originally Posted by HallsofIvy View Post
    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.
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    The points are distinct and collinear for a=3: \;\;(3,1),~(1,3),~(2,2).
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