Can there be 3 collinear points in a curve?

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Discussion Overview

The discussion revolves around the possibility of having three collinear points on a curve, specifically focusing on quadratic curves. Participants explore the implications of tangent lines, the nature of curves, and the mathematical definitions involved in determining collinearity.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that as one picks smaller and smaller values between two x-values, a curve appears to become straighter, leading to the question of whether three collinear points can exist on a quadratic curve.
  • Others argue that a quadratic function, defined as y = ax² + bx + c, can have at most two distinct solutions when set equal to a linear function, implying that three collinear points cannot exist on a quadratic curve.
  • A participant mentions that while three collinear points can exist on some curves, this does not apply to quadratic curves.
  • There is a discussion about the nature of tangent lines, with some asserting that a tangent line only touches a curve at one point, while others question this definition and explore the implications of infinitely small points.
  • Some participants express confusion regarding the concept of points and tangents, suggesting that the infinite divisibility of points might imply more complex interactions between tangent lines and curves.
  • One participant provides a specific example involving the tangent line to y = x² at (0, 0), illustrating that while the tangent touches at one point, it does not actually touch at nearby points, reinforcing the idea that tangents do not intersect curves at more than one point.

Areas of Agreement / Disagreement

Participants generally disagree on the possibility of having three collinear points on a quadratic curve, with some asserting it is impossible and others suggesting that the nature of curves allows for such configurations under certain conditions. The discussion remains unresolved regarding the implications of tangent lines and the definition of points.

Contextual Notes

Participants express various assumptions about the nature of curves and points, including the concept of smoothness and the behavior of tangent lines. There is a lack of consensus on the definitions and implications of these mathematical concepts.

Skhandelwal
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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 that's 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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Skhandelwal said:
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 that's true, don't you think it is possible for 3 collinear points to be there in a curve?
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:
Here, I'll be more specific, in a quadratic curve.
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? :)
 
VietDao29 said:
But that's a quadratic equation, and so, there are at most 2 distinct solutions. There cannot be 3.
Can you get this? :)




Err... I mean that I agree with VietDao29 that quadratic equations can't have three colinear points.
 
can there be three common solutions of a luinear and a quadratic equation?
 
No, because 3 common solutions for both a linear and quadratic equation implies, by definition, that the quadratic equation has 3 colinear points.
 
Skhandelwal said:
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 that's 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.

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?
 
I was just trying to say that are curves made up of smaller lines?
 
Skhandelwal said:
I was just trying to say that are curves made up of smaller lines?

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.
 
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.
 
  • #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?
 
  • #11
Skhandelwal said:
ohh, so you mean that tangent line only touches 1 point of the curve?

That's pretty much how a tangent line is defined.

Skhandelwal said:
But how is that possible b/c points can always get smaller?

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.

Skhandelwal said:
Or are points as specific as zero repeating 1 after a decimal?

I don't know what you mean by this.
 
  • #12
d_leet said:
That's pretty much how a tangent line is defined.

Be careful there; this is often false.
 
  • #13
CRGreathouse said:
Be careful there; this is often false.

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.
 
  • #14
d_leet said:
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.

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.
 
  • #15
What I don't get is that if tangent point only touches 1 point and a point can't be defined because it always gets smaller. For instance, the point tangent line touches, can't 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 isn't 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.
 
  • #16
Skhandelwal said:
What I don't get is that if tangent point only touches 1 point and a point can't be defined because it always gets smaller. For instance, the point tangent line touches, can't 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 isn't 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.

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.
 
  • #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)
 
  • #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.
 
  • #19
Skhandelwal said:
What I don't get is that if tangent point only touches 1 point and a point can't be defined because it always gets smaller. For instance, the point tangent line touches, can't 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 isn't 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.

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.
 
  • #20
Skhandelwal said:
What I don't get is that if tangent point only touches 1 point and a point can't be defined because it always gets smaller.
Yes, a point can be and is defined. It doesn't always get smaller. A point is a point- it has no size.

Skhandelwal said:
So you are saying that when a tangent line touches, it is as specific as the smallest point possible?(ex. .000000000000...till infinity 1)
There is no such thing as "the smallest point possible". A point is a point- it has no size.
 
Last edited by a moderator:
  • #21
no smallest point huh, then tell me a point b/w .000000...till infinity..1 and 0.
 
  • #22
Skhandelwal said:
no smallest point huh, then tell me a point b/w .000000...till infinity..1 and 0.

This:

".0000...till infinity...1"

is complete nonsense. It's not a real number, it's some symbols you've put together and hoped they had some meaning. They don't.


There is no smallest positive real number. If x>0 then x>x/2>0, so we've just found a smaller positive real number.

edit-I'm being generous and interpreting "smallest point" to mean "smallest real number" in some form. "smallest point" as if different points somehow have a different size or dimensions is a meaningless interpretation.
 
  • #23
shmoe said:
This:

".0000...till infinity...1"

is complete nonsense. It's not a real number, it's some symbols you've put together and hoped they had some meaning. They don't.

I read that the way MoD does above:

".0000...till infinity...1" = [tex]\lim_{n\rightarrow\infty}10^{-n}=0[/tex]

As such, there is no point strictly between ".0000...till infinity...1" and 0 because they are the same number.
 
  • #24
CRGreathouse said:
I read that the way MoD does above:

".0000...till infinity...1" = [tex]\lim_{n\rightarrow\infty}10^{-n}=0[/tex]

As such, there is no point strictly between ".0000...till infinity...1" and 0 because they are the same number.

I read it as total nonsense, attempting to put a 1 at "the end". What else is the 1 there for, except as a conceptual misunderstanding?
 
  • #25
shmoe said:
I read it as total nonsense, attempting to put a 1 at "the end". What else is the 1 there for, except as a conceptual misunderstanding?

It's a conceptual misunderstanding, but this is a way to think about it in a rigorous sense. The misunderstanding is then changed from your "just playing with symbols" to the assumption that the two are different just because they're written differently (or that there's 'somewhere to go' after infinity).
 
  • #26
I just got it, thanks a lot.
 

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