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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. :)

  • 2 years ago
  • 2 years ago

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  1. ilikephysics2 Group Title
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    Its true...i didnt look it up

    • 2 years ago
  2. klimenkov Group Title
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    It is TRUE.

    • 2 years ago
  3. myininaya Group Title
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    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?

    • 2 years ago
  4. ilikephysics2 Group Title
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    that is the formula for it , look in your book

    • 2 years ago
  5. experimentX Group Title
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    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.

    • 2 years ago
  6. ilikephysics2 Group Title
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    @experimentX its the formula for it man

    • 2 years ago
  7. experimentX Group Title
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    formula is just expression of logic.

    • 2 years ago
  8. ilikephysics2 Group Title
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    yeah

    • 2 years ago
  9. CliffSedge Group Title
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    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.

    • 2 years ago
  10. CliffSedge Group Title
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    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.

    • 2 years ago
  11. myininaya Group Title
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    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.

    • 2 years ago
  12. myininaya Group Title
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    I left out the word "or"

    • 2 years ago
  13. myininaya Group Title
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    or an ellipse

    • 2 years ago
  14. CliffSedge Group Title
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    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.

    • 2 years ago
  15. CliffSedge Group Title
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    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.

    • 2 years ago
  16. Zarkon Group Title
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    if you let r=0 then you will destroy properties that all 'normal' circles have

    • 2 years ago
  17. myininaya Group Title
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    You put quotation marks around normal because you do see it as a circle @zarkon , but not a "normal" circle ?

    • 2 years ago
  18. Zarkon Group Title
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    I would call it a degenerate circle

    • 2 years ago
  19. CliffSedge Group Title
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    I like "degenerate."

    • 2 years ago
  20. CliffSedge Group Title
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    It's a shorter way of saying "circle with radius zero."

    • 2 years ago
  21. experimentX Group Title
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    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

    • 2 years ago
  22. precal Group Title
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    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.......

    • 2 years ago
  23. CliffSedge Group Title
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    I don't see why the center and circumference cannot coincide. There is nothing in the definition of a circle that forbids that.

    • 2 years ago
  24. precal Group Title
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    true so true........

    • 2 years ago
  25. Zarkon Group Title
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    that depends on what definition you use

    • 2 years ago
  26. Zarkon Group Title
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    I like my circles to have interiors..and the break the plane into two regions (not including the circle itself)

    • 2 years ago
  27. Zarkon Group Title
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    that way all my theorems hold and make sense

    • 2 years ago
  28. joemath314159 Group Title
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    Maybe thinking of the equation as a "circle" is what makes it seem confusing. Think of it as a distance function:\[d: \mathbb{R}^2\times\mathbb{R}^2\rightarrow\mathbb{R}\]\[((x_1,y_1),(x_2,y_2))\longmapsto \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\]

    • 2 years ago
  29. joemath314159 Group Title
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    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."

    • 2 years ago
  30. CliffSedge Group Title
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    @joemath314159 the equation of a circle, the distance formula, and the pythagorean theorem are all the same thing.

    • 2 years ago
  31. Zarkon Group Title
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    draw a tangent to a circle with radius zero

    • 2 years ago
  32. CliffSedge Group Title
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    @Zarkon why does a tangent need to be defined in order for it to be a circle?

    • 2 years ago
  33. Zarkon Group Title
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    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"

    • 2 years ago
  34. CliffSedge Group Title
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    And even if there is no single unique tangent to that point, a tangent can still be drawn.

    • 2 years ago
  35. CliffSedge Group Title
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    Those are properties of some circles, but are they included in the definition of "circle?"

    • 2 years ago
  36. CliffSedge Group Title
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    Argument from personal preference isn't valid, I'd think.

    • 2 years ago
  37. Zarkon Group Title
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    most are theorems that require that a circle have an interior

    • 2 years ago
  38. CliffSedge Group Title
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    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."

    • 2 years ago
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