## myininaya Group Title True/False: A graph that contains just a single point (h,k) can be written as (x-h)^2+(y-k)^2=0 which means it can be seen as a circle with radius 0. Please state what you think. Don't look it up on a website because I can do that if that is what I wanted. :) one year ago one year ago

1. ilikephysics2 Group Title

Its true...i didnt look it up

2. klimenkov Group Title

It is TRUE.

3. myininaya Group Title

That is what I say to but people actually do take the other side on this one. If there is any reason why you think it is true, can you say why?

4. ilikephysics2 Group Title

that is the formula for it , look in your book

5. experimentX Group Title

the equality only hold for (x,y) = (h,k) for any set of point other than (h, k) this relation is not valid in Real plane.

6. ilikephysics2 Group Title

@experimentX its the formula for it man

7. experimentX Group Title

formula is just expression of logic.

8. ilikephysics2 Group Title

yeah

9. CliffSedge Group Title

Some people don't like the idea of calling something with zero size a circle (or any other plane figure), but I don't have a problem with it. If you want to be more precise, it's the limit of a circle as its radius approaches zero.

10. CliffSedge Group Title

That is the standard form of an equation for a circle with r=0, so why not? I would be more general and say it's an ellipse of size zero, but I'm a dork like that.

11. myininaya Group Title

I think it depends on how you define a circle. I would say the radius could be greater than equal to 0. Someone told me you can actually prove that when you have (x-h)^2+(y-k)^2=0 this is a circle.

12. myininaya Group Title

I left out the word "or"

13. myininaya Group Title

or an ellipse

14. CliffSedge Group Title

Depends on if you want a synthetic geometry definition, analytic geometry definition, calculus definition. As far as I'm concerned, It is a circle. It's a circle with r=0.

15. CliffSedge Group Title

So, in more plain language (or using a geometry definition), it would be better to call it a point rather than a circle, but as long as you are clear in your definitions and can show your logic is consistent in either case, either way works.

16. Zarkon Group Title

if you let r=0 then you will destroy properties that all 'normal' circles have

17. myininaya Group Title

You put quotation marks around normal because you do see it as a circle @zarkon , but not a "normal" circle ?

18. Zarkon Group Title

I would call it a degenerate circle

19. CliffSedge Group Title

I like "degenerate."

20. CliffSedge Group Title

It's a shorter way of saying "circle with radius zero."

21. experimentX Group Title

perhaps that way we could avoid confusing the difference between these two. (x-h)^2+(y-k)^2=0 a(x-h)^2+b(y-k)^2=0

22. precal Group Title

but isn't a circle just defined as a center with all points on the circumference equadistant fro the center. Hmmmmm center and circumference are the same here.......

23. CliffSedge Group Title

I don't see why the center and circumference cannot coincide. There is nothing in the definition of a circle that forbids that.

24. precal Group Title

true so true........

25. Zarkon Group Title

that depends on what definition you use

26. Zarkon Group Title

I like my circles to have interiors..and the break the plane into two regions (not including the circle itself)

27. Zarkon Group Title

that way all my theorems hold and make sense

28. joemath314159 Group Title

Maybe thinking of the equation as a "circle" is what makes it seem confusing. Think of it as a distance function:$d: \mathbb{R}^2\times\mathbb{R}^2\rightarrow\mathbb{R}$$((x_1,y_1),(x_2,y_2))\longmapsto \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

29. joemath314159 Group Title

Then in a sense, when you have:$(x-h)^2+(y-k)^2=0$you are saying "I want the set of all points (x,y) such that the distance between (x,y) and (h,k) is zero."

30. CliffSedge Group Title

@joemath314159 the equation of a circle, the distance formula, and the pythagorean theorem are all the same thing.

31. Zarkon Group Title

draw a tangent to a circle with radius zero

32. CliffSedge Group Title

@Zarkon why does a tangent need to be defined in order for it to be a circle?

33. Zarkon Group Title

all other circles have that property..and many other properties that are destroyed by having r=0...maybe it should have its own name...like "point"

34. CliffSedge Group Title

And even if there is no single unique tangent to that point, a tangent can still be drawn.

35. CliffSedge Group Title

Those are properties of some circles, but are they included in the definition of "circle?"

36. CliffSedge Group Title

Argument from personal preference isn't valid, I'd think.

37. Zarkon Group Title

most are theorems that require that a circle have an interior

38. CliffSedge Group Title

Then that's just too bad. You use those theorems when they apply. If they don't apply, you don't use the theorems. Whether or not particular theorems apply does not change the definition of "circle."