Can there be 3 collinear points in a curve?

[Skhandelwal]

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.

 

[VietDao29]

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.

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? :)

 

[CRGreathouse]

But that's a quadratic equation, and so, there are at most 2 distinct solutions. There cannot be 3.
Can you get this? :)

GAUSS'd!​

Err... I mean that I agree with VietDao29 that quadratic equations can't have three colinear points.

 

[mathwonk]

can there be three common solutions of a luinear and a quadratic equation?

 

[Nebosuke]

No, because 3 common solutions for both a linear and quadratic equation implies, by definition, that the quadratic equation has 3 colinear points.

 

[HallsofIvy]

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?

 

[Skhandelwal]

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

 

[CRGreathouse]

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.

 

[JonF]

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.

 

[Skhandelwal]

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?

 

[d_leet]

ohh, so you mean that tangent line only touches 1 point of the curve?

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

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.

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

I don't know what you mean by this.

 

[CRGreathouse]

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

Be careful there; this is often false.

 

[d_leet]

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.

 

[CRGreathouse]

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.

 

[Skhandelwal]

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.

 

[CRGreathouse]

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.

 

[Skhandelwal]

So you are saying that when a tangent line touches, it is as specific as the smallest point possible?(ex. .000000000000...till infinity 1)

 

[Moo Of Doom]

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.

 

[d_leet]

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.

 

[HallsofIvy]

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.

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.

 

[Skhandelwal]

no smallest point huh, then tell me a point b/w .000000...till infinity..1 and 0.

 

[shmoe]

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 postive 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.

 

[CRGreathouse]

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" = $$\lim_{n\rightarrow\infty}10^{-n}=0$$

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

 

[shmoe]

I read that the way MoD does above:

".0000...till infinity...1" = $$\lim_{n\rightarrow\infty}10^{-n}=0$$

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?

 

[CRGreathouse]

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).

 

[Skhandelwal]

I just got it, thanks a lot.