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Math Help - Semiaxes of an ellipse (3D)

  1. #1
    MHF Contributor alexmahone's Avatar
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    Semiaxes of an ellipse (3D)

    Find the semiaxes of the ellipse in which the plane z=x meets the cylinder x^2+y^2=4.
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    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.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    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
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  4. #4
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by FernandoRevilla View Post
    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.
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by alexmahone View Post
    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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