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Circle radius 0, algebraic manipulation

  1. Apr 30, 2012 #1
    1. The problem statement, all variables and given/known data

    Going over some old tests, I am asked to find the contour of the function:

    T = 100 - x^2 - y^2

    at T = 100, T = 0, etc.

    I have a question regarding the contour at T = 100

    2. Relevant equations



    3. The attempt at a solution

    Consider T = 100

    100 = 100 - x^2 - y^2
    0 = -x^2 - y^2

    This is a circle with radius 0. I knew at the time that this must be the top of the paraboloid. However, I also noted that

    0 = -x^2 -y^2
    0 = x^2 + y^2
    y^2 = -x^2
    y = sqrt(-x^2)

    Is it correct to determine this result as "there exists a real solution only at x = 0, thus the only point at this contour is (0,0)"?
     
  2. jcsd
  3. Apr 30, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    In fact, [tex]100- x^2- y^2= T[/tex] is the same as [tex]x^2+ y^2= 100- T[/tex] which, for T< 100, is the equation of a circle, with center at (0, 0) and radius [itex]\sqrt{100- T}[/itex]. When T= 100, that reduces to [tex]x^2+ y^2= 0[/tex] which is, as you say, only true for (x, y)= (0, 0). The contour is a single point (which you can think of as a circle with radius "0").
     
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