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Parametric Equations of an ellipse

  1. Mar 23, 2005 #1
    The ellipse [tex]\frac{x^2}{3^2} + \frac{y^2}{4^2} = 1[/tex]
    can be drawn with parametric equations. Assume the curve is traced clockwise as the parameter increases.

    If [tex] x=3cos(t)[/tex]

    then y = ___________________________


    wouldnt i just sub x into the ellipse equation and solve for y?

    well i did that and got [tex]\sqrt{(-1/16*((3*cos(t))^2/9)+1)}[/tex]

    but there's a negative sign inside the sqrt function, so it's not possible
     
  2. jcsd
  3. Mar 23, 2005 #2
    [tex]\sqrt{(-1/16*((3*cos(t))^2/9)+1)}[/tex]

    [tex]\sqrt{(-1/16*(9cos^2(t)/9)+1)}[/tex]

    [tex]\sqrt{(-cos^2(t)/16+16/16)}[/tex]

    [tex]\sqrt{\frac{(16-cos^2(t))}{16}} [/tex]

    [tex]\frac{\sqrt{16-cos^2(t)}}{4} [/tex]

    [tex]\frac{\sqrt{(4-cos(t))(4+cos(t))}}{4} [/tex]


    Im sure that can simplify more, but I'm out of ideas.
     
    Last edited: Mar 23, 2005
  4. Mar 23, 2005 #3
    Also consider that a circle is an ellipse with a = b = 1, in which case the parametric equations are:

    [tex] x(t) = a cos(t) = cos(t) [/tex]
    [tex] y(t) = b sin(t) = sin(t) [/tex]
     
  5. Mar 23, 2005 #4

    dextercioby

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    Okay.I think it's not too difficult to show that
    [tex] y=4\sin t [/tex]

    Daniel.
     
  6. Mar 23, 2005 #5
    [tex]\frac{\sqrt{(4-cos(t))(4+cos(t))}}{4} [/tex]

    and y = 4*sin(t) is incorrect. I really get and understand how you got 4*sin(t). but anyone know why these answers are incorrect?
     
  7. Mar 23, 2005 #6

    dextercioby

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    [tex] \frac{y^{2}}{16}=1-\cos^{2}t=\sin^{2}t\Rightarrow y^{2}=(4\sin t)^{2}\Rightarrow y=\pm 4\sin t [/tex]...U can choose the "-" sign ([tex] y\searrow \ \mbox{when} \ t\nearrow [/tex])...

    Daniel.
     
  8. Mar 23, 2005 #7
    The answer would be [tex] y = -4sin(t) [/tex] because the particle moves clockwise, and as [tex] t \nearrow, sin(t) \mbox { travels counter clockwise.} [/tex]

    For [tex] sin(t) \mbox{ to travel clockwise you would need to multiply the parameter by -1} [/tex]

    [tex] y(t) = 4sin(-t) \mbox{ which equals } y(t) = -4sin(t) \mbox{ by properties of the sin function} [/tex]
     
  9. Mar 23, 2005 #8

    dextercioby

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    Well,what do you know,it's the same thing with what i've written...:tongue2:

    Daniel.
     
  10. Mar 23, 2005 #9
    I was explaining to him why :)
     
  11. Mar 23, 2005 #10

    dextercioby

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    Sure you were...:wink: However,i still think the OP needs to do some thinking on this problem.

    Daniel.
     
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