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The ellipse is: $ x(t)=a \cos(wt-c)\\ y(t)=b \cos(wt-d) $

What are:

  1. major axis length
  2. minor axis length
  3. angle of major axis with $x$ axis?
  4. the parametric form ? $(ax^2+by^2+cxy+dx+fy+...=g)$
  • 3
    I think it is quite unpolite to order things instead of asking them. Moreover you should say why you are interested in this problem (homework maybe?) and what have you tried. Then we will be all glad to help you.2012-09-26

2 Answers 2

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Expand $\cos(wt-c)$ and $\cos(wt-d)$ using $\cos(A-B)=\cos A\cos B+\sin A\sin B$

Solve for $\cos(wt), \sin(wt)$

Use $\cos^2(wt)+ \sin^2(wt)=1$ to remove $wt$ from the given equations to get

$x^2b^2+y^2a^2-2xyab\cos(c-d)-a^2b^2sin^2(c+d)=0$

Use Rotation of axes, to remove $xy$ term from the equation to get the standard form $\frac{X^2}{A^2}+\frac{Y^2}{B^2}=1$.

The major axis length= 2max$(A,B)$

The minor axis length= 2min$(A,B)$

The parametric form would be $(A\cos \alpha, B\sin \alpha)$

  • 0
    @Ali, how have you calculated the question#3? I think, it should be the angle of rotation $\theta$ if X is the new major axis, and $\frac{\pi}{2}+ \theta$ if Y is the new major axis.2012-09-27
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  1. a
  2. b
  3. $(3\pi/2)-d+c$
  4. change cos to sine by adding $3\pi/2$ and do mainpulation.
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    Dear Babak, I wanted to change the form to parametric by calculating x^2+y^2 to omitting cos(wt). But I couldn't.2012-09-26