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Re: Conic sections when we get in Geometry // Teaching True Mathematics

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Archimedes Plutonium <plutonium.archimedes@gmail.com> writes:

>In High School when we get to geometry and conic sections we learn 
>ellipse is a cylinder section never a conic and what we thought was a 
>conic ellipse turns out to be an oval.

Teacher, Teacher, can we study this proof, please please please???

Some preliminaries:

Top view of the conic section and depiction of the coordinate system used 
in the proof:

              ^ x
              |
             -+- <= x=h
         .'   |   `.
        .     |     .
        |     |     |
        '     |     '      
         `.   |   .'
 y <----------+ <= x=0
             
Cone (side view):
                 .
                /|\
               / | \
              /b |  \
             /---+---' <= x = h
            /    |'   \
           /   ' |     \
          / '    |      \
x = 0 => '-------+-------\
        /    a   |        \

Proof:

r(x) = a - ((a-b)/h)x  and  d(x) = a - ((a+b)/h)x,  hence

y(x)^2 = r(x)^2 - d(x)^2 = ab - ab(2x/h - 1)^2 = ab(1 - 4(x - h/2)^2/h^2.

Hence (1/ab)y(x)^2 + (4/h^2)(x - h/2)^2 = 1  ...equation of an ellipse

qed

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#2-30 & 3-1, Sigma error and Geometry Logic// Teaching True Mathematics Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-05-14 14:21 -0700
  Conic sections when we get in Geometry // Teaching True Mathematics Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-05-15 01:16 -0700
    Re: Conic sections when we get in Geometry // Teaching True Mathematics moroney@world.std.spaamtrap.com (Michael Moroney) - 2018-05-15 11:19 +0000
      Re: Conic sections when we get in Geometry // Teaching True Mathematics moroney@world.std.spaamtrap.com (Michael Moroney) - 2018-05-15 19:15 +0000
        Jacob Barandes, Marty Walsh, Charlie Baker, Howard Berg, Michael Brenner,   -- Moroney asks why you keep failing with identity of Real Electron = muon, proton=840MeV, .5MeV = Dirac's monopole Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-05-15 12:36 -0700
          Re: Archimedes "Failure" Plutonium flunked the Math Test of a lifetime-generation test moroney@world.std.spaamtrap.com (Michael Moroney) - 2018-05-15 20:39 +0000
    we have the Cone tube, and the Cylinder tube, now we have the Ellipse tube // Teaching True Mathematics Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-05-15 11:28 -0700
      Re: we have the Cone tube, and the Cylinder tube, now we have the Ellipse tube // Teaching True Mathematics Alan Mackenzie <acm@muc.de> - 2018-05-17 18:25 +0000

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