Semiaxes of an ellipse (3D)

**alexmahone** · January 1st, 2011, 07:53 AM

Find the semiaxes of the ellipse in which the plane $z=x$ meets the cylinder $x^2+y^2=4$ .

**tom@ballooncalculus** · January 1st, 2011, 08:25 AM

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.

**FernandoRevilla** · January 1st, 2011, 10:02 AM

The parametric equations of the cylinder are:

$C \equiv\begin{Bmatrix}x=2\cos t\\y=2\sin t\\z=2\cos t\end{matrix}\quad (t\in [0,2\pi])$

The square of the distance from the center $(0,0,0)$ of the ellipse to any point of $C$ is:

$d^2(x,y,z)=\ldots=4+4\cos^2 t$

Easily you can find the maximum $M$ and the minimum $m$ absolute, of $d^2$ . The semi-axes are $\sqrt{M}$ and $\sqrt{m}$ .

Fernando Revilla

**alexmahone** · January 3rd, 2011, 07:58 AM

Originally Posted by FernandoRevilla

The parametric equations of the cylinder are:

$C \equiv\begin{Bmatrix}x=2\cos t\\y=2\sin t\\z=2\cos t\end{matrix}\quad (t\in [0,2\pi])$

The square of the distance from the center $(0,0,0)$ of the ellipse to any point of $C$ is:

$d^2(x,y,z)=\ldots=4+4\cos^2 t$

Easily you can find the maximum $M$ and the minimum $m$ absolute, of $d^2$ . The semi-axes are $\sqrt{M}$ and $\sqrt{m}$ .

Fernando Revilla

I think you meant 'parametric equations of the ellipse'.

Using your method, I get the semi-major axis as $2\sqrt{2}$ and semi-minor axis as $2$ .

**FernandoRevilla** · January 3rd, 2011, 08:50 AM

Originally Posted by alexmahone

I think you meant 'parametric equations of the ellipse'.

Yes, of course I meant ellipse.

Using your method, I get the semi-major axis as $2\sqrt{2}$ and semi-minor axis as $2$ .

Right.

Fernando Revilla

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