Parametric Equations of an ellipse

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  • #1
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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
 

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  • #2
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[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.
 
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  • #3
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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]
 
  • #4
dextercioby
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Okay.I think it's not too difficult to show that
[tex] y=4\sin t [/tex]

Daniel.
 
  • #5
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[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?
 
  • #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.
 
  • #7
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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]
 
  • #8
dextercioby
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Well,what do you know,it's the same thing with what i've written...:tongue2:

Daniel.
 
  • #9
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I was explaining to him why :)
 
  • #10
dextercioby
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whozum said:
Im sure that can simplify more, but I'm out of ideas.

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

Daniel.
 

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