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Can there be 3 collinear points in a curve?

  1. Aug 30, 2006 #1
    I was wondering, since there is no limit to how many points you can pick b/w x value 1 to 2. If you keep picking smaller and smaller values, the curve keeps getting more and more straight. It is similar to how .000000000...till infinity then 1 equals zero. So if thats true, don't you think it is possible for 3 collinear points to be there in a curve? Here, I'll be more specific, in a quadratic curve.
     
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  3. Aug 30, 2006 #2

    VietDao29

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    Yes, of course, it is possible, say you have a curve y = x3, the 3 point (-1, -1), (0, 0), and (1, 1) are collinear, and on the curve. :smile:
    No there cannot be more than, or equal to 3 collinear points the lie on a quardratic curve. Why?
    The function of a quadratic curve is: y = ax2 + bx + c (a not 0)
    And the function of a line is y = dx + e
    Now if there are 3 points that both lie on the curse, and the line mentioned above, then there must be 3 distinct solutions to the equation:

    ax2 + bx + c = dx + e
    <=> ax2 + (b - d)x + (c - e) = 0
    But that's a quadratic equation, and so, there are at most 2 distinct solutions. There cannot be 3.
    Can you get this? :)
     
  4. Aug 30, 2006 #3

    CRGreathouse

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    Err... I mean that I agree with VietDao29 that quadratic equations can't have three colinear points.
     
  5. Aug 30, 2006 #4

    mathwonk

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    can there be three common solutions of a luinear and a quadratic equation?
     
  6. Sep 1, 2006 #5
    No, because 3 common solutions for both a linear and quadratic equation implies, by definition, that the quadratic equation has 3 colinear points.
     
  7. Sep 2, 2006 #6

    HallsofIvy

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    Of course, there can be 3 collinear points on some curves, not on others. But that has nothing to do with "picking smaller and smaller values" or the curve "getting more and more straight". What are you really trying to say?
     
  8. Sep 2, 2006 #7
    I was just trying to say that are curves made up of smaller lines?
     
  9. Sep 2, 2006 #8

    CRGreathouse

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    Sure, if the curves are linear.

    You seem to be working off the idea of smoothness implying that a curve is "arbitrarily close" to linear with enough magnification. If you make that concrete with a definition of how close you'd like a curve to be to having three points on a line, it would be simple to show an infinite family of quadratic curves meeting your definition. But to actually have three points exactly on the curve, it can't have degree 2 if the points are colinear.
     
  10. Sep 3, 2006 #9
    Uh I think the OP is asking about for any given differentiable curve whether we can find 3 collinear points.. His reasoning is that the tangent line seems to approximate the curve for a sufficiently small region.

    The answer is no. There are plenty of curves where this fails. Such as y=x^2. Just because the tangent line gets really close to a small collection of points on our curve doesn’t mean they are actually on the curve.
     
  11. Sep 3, 2006 #10
    ohh, so you mean that tangent line only touches 1 point of the curve? But how is that possible b/c points can always get smaller? Or are points as specific as zero repeating 1 after a decimal?
     
  12. Sep 3, 2006 #11
    That's pretty much how a tangent line is defined.

    There are an infinite number of points between any two points, but that doesn't imply that a tangent line will touch more than a single point.

    I don't know what you mean by this.
     
  13. Sep 3, 2006 #12

    CRGreathouse

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    Be careful there; this is often false.
     
  14. Sep 3, 2006 #13
    Yes, I know that. It's not true in the case of y=x3, but for y=x2 it will be true that the tangent line intersects/touches the graph at only one point.
     
  15. Sep 3, 2006 #14

    CRGreathouse

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    I know *you* knew, but since there are some neophytes on this thread I wanted to point that out. You're quite right on x^2=y, of course.
     
  16. Sep 4, 2006 #15
    What I dont get is that if tangent point only touches 1 point and a point cant be defined because it always gets smaller. For instance, the point tangent line touches, cant there be more points in that same space because points can always get smaller. As an example. A line touches the x value 3 on a curve. but isnt it possible that while touching three, is has also touched 3.000000000000...infinity 1 and 3.00000000..infinity 2? Perhaps these examples are too extreme because technically, that would equal 3. But may be 3.000000.. till a million, then 1. All I am saying is that if it touches a point(that has some sort of region), it can always be devided into smaller points. Hope I made myself clear.
     
  17. Sep 4, 2006 #16

    CRGreathouse

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    Let's consider the tangent line y = 0 to the function y = x^2 at (0, 0). Certainly they touch at (0, 0), but let's consider x = 10^{-1,000,000}. The tanget line has y = 0 there, and the function has y = 10^{-2,000,000}. Now that's really close to touching, but it's not actually touching. A similar argumen can be made for all points other than at x = 0, where they do touch.
     
  18. Sep 4, 2006 #17
    So you are saying that when a tangent line touches, it is as specific as the smallest point possible?(ex. .000000000000...till infinity 1)
     
  19. Sep 4, 2006 #18
    0.0000000..."till infinity" 1 = 0 because if you have a string of infinite zeroes, you can't put a 1 at the end because there is no end.

    All points have 0 length, 0 area, and 0 volume. You can't divide them into even smaller points. Any region that includes more than one point can be divided as often as you like into smaller and smaller regions, but each region will always contain an infinite number of points.
     
  20. Sep 4, 2006 #19
    If those two points do not lie on a line on the curve, then the tangent line will not touch more than one of them.
     
  21. Sep 4, 2006 #20

    HallsofIvy

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    Yes, a point can be and is defined. It doesn't always get smaller. A point is a point- it has no size.

    There is no such thing as "the smallest point possible". A point is a point- it has no size.
     
    Last edited: Sep 5, 2006
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