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Cylindrical section is an ellipse?

  1. Jul 23, 2008 #1
    Prove or disprove: The intersection of the plane [tex]x+y+z=1[/tex] and the cylinder [tex]x^2+y^2=1[/tex] is an ellipse.
     
  2. jcsd
  3. Jul 23, 2008 #2
    If you perform a change of variables so that the x+y+z=1 plane is the new 'x-y plane' then do you not get a new equation for your cylinder? You can then work it out by setting the new 'z' to zero?
     
  4. Jul 24, 2008 #3

    HallsofIvy

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    The intersection of the two figures is the values of (x, y, z) that satisfy both equations simutaneously so solve the two equations simultaneously: From the first equation, y= 1- x- z. What do you get when you replace y by that in your second equation?
     
  5. Jul 24, 2008 #4
    Thanks for the help... I actually found a (more general) proof in a geometry textbook (Geometry and the Imagination by David Hilbert)
     
  6. Jul 25, 2008 #5
    It would have to be an ellipse because the plane isn't parallel to the cylinder. The normal vector is i+j+k which makes an angle of [tex]cos\alpha =\frac{1}{\sqrt3}[/tex] with all three axes.
     
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