Find the semiaxes of the ellipse in which the plane $\displaystyle z=x$ meets the cylinder $\displaystyle x^2+y^2=4$.
Have you visualised the plane and the cylinder? And their intersection? Then check semi axes at Ellipse - Wikipedia, the free encyclopedia and just find how far the intersection is from the origin, along the y axis, and along the line z = x in the x,z plane.
If visualising is the problem, do say.
The parametric equations of the cylinder are:
$\displaystyle C \equiv\begin{Bmatrix}x=2\cos t\\y=2\sin t\\z=2\cos t\end{matrix}\quad (t\in [0,2\pi]) $
The square of the distance from the center $\displaystyle (0,0,0)$ of the ellipse to any point of $\displaystyle C$ is:
$\displaystyle d^2(x,y,z)=\ldots=4+4\cos^2 t$
Easily you can find the maximum $\displaystyle M$ and the minimum $\displaystyle m$ absolute, of $\displaystyle d^2$ . The semi-axes are $\displaystyle \sqrt{M}$ and $\displaystyle \sqrt{m}$ .
Fernando Revilla
Yes, of course I meant ellipse.
Using your method, I get the semi-major axis as $\displaystyle 2\sqrt{2}$ and semi-minor axis as $\displaystyle 2$.
Right.
Fernando Revilla