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

Question

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 ? $(ax^2+by^2+cxy+dx+fy+...=g)$

Answer 1

1. a
2. b
3. $(3\pi/2)-d+c$
4. change cos to sine by adding $3\pi/2$ and do mainpulation.

Answer 2

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