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Do these 3 systems of equations all all define the same curve?

  1. Dec 18, 2011 #1
    1. The problem statement, all variables and given/known data

    Consider three systems of equations:

    x^2 + y^2 + z^2 = 1
    y^2 + z^2 = 1

    x^2 + y^2 + z^2 = 1
    x = 0

    y^2 + z^2 = 1
    x = 0

    Which of these define the same curve and which define different ones?

    2. Relevant equations

    x^2 + y^2 + z^2 = R^2 is a sphere
    x,y, or z = # is a plane
    (x,y,z)^2 + (x,y,z)^2 = # is a cylinder

    3. The attempt at a solution

    I think they all define the same curve; here is why...

    Equation 1 of the first system is a sphere centered at the origin with a radius of 1.
    Equation 2 of the first system is a cylinder centered around the x axis.
    The curve represented by those two equations is the points where both equations are satisfied; AKA: where they intersect. They intersect at x=0 in a circle given by y^2+z^2 = 1 .

    Equation 1 of the second system is a sphere centered at the origin with a radius of 1.
    Equation 2 of the second system is the yz plane at x=0.
    The curve represented by those two equations is the points where both equations are satisfied; AKA: where they intersect. They intersect at x=0 in a circle given by y^2+z^2 = 1 .


    Equation 1 of the third system is a cylinder centered at the x axis with a radius of 1.
    Equation 2 of the third system is the yz plane at x=0.
    The curve represented by those two equations is the points where both equations are satisfied; AKA: where they intersect. They intersect at x=0 in a circle given by y^2+z^2 = 1 .




    Am I right?:biggrin:
     
  2. jcsd
  3. Dec 18, 2011 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, you are right.
     
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