# major and minor axis of ellipse, $\phi$ (degree from $x$ axis)

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 ? $(ax^2+by^2+cxy+dx+fy+...=g)$
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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. – uforoboa Sep 26 '12 at 13:49

## 2 Answers

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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Hi. I didn't want to order. I wanted to write the main part and not time consuming letters. I need it for rotordynamic analysis. When I solve a harmonic response, I get displacement in x and y direction in that format. I want to find pick response and the phase. Pick response is major axis length and phase is the angle between mahjor axis with x axis. It is not a school homework. Would you please help me in this regard? – Ali Sep 26 '12 at 14:01
Dear Babak, I wanted to change the form to parametric by calculating x^2+y^2 to omitting cos(wt). But I couldn't. – Ali Sep 26 '12 at 14:04

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 $(A\cos \alpha, B\sin \alpha)$

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I used b^2 x^2+a^2 y^2-2xyabcos(c-d).... I know that cos (2 alpha)= (b^2-a^2)/(-2abcos(c-d). But for b=5,a=10,c=30,d=90 we don't have a alpha.cos2alpha=1.5 . But Alpha should be 16.845 degree. – Ali Sep 26 '12 at 14:23
That is Cot (2*alpha)= .... and not cos . Thank you "lab bhattacharjee". My question is solved. – Ali Sep 26 '12 at 14:56
@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. – lab bhattacharjee Sep 27 '12 at 9:47