Results 1 to 4 of 4

Math Help - length of large and small axis of the ellipse

  1. #1
    Newbie
    Joined
    Mar 2013
    From
    Ljubljana
    Posts
    2

    length of large and small axis of the ellipse

    I need to calculate this elipse:


    $c^2=4x_1^2+3x_2^2-2\sqrt2{x_1x_2}$

    1. c2=1
    2. c2=4


    I need to calculate direction and the length of large and small axis of the ellipse.


    (hint: own vector and Eigenvalues)

    Thanks for your help.


    Last edited by mikeno80; March 21st 2013 at 10:17 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,612
    Thanks
    591

    Re: length of large and small axis of the ellipse

    Hey mikeno80.

    This looks like a problem involving quadratic forms:

    Principal axis theorem - Wikipedia, the free encyclopedia
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,546
    Thanks
    1395

    Re: length of large and small axis of the ellipse

    One method: if we rotate and xy-coordinate system through angle \theta, we have new coordinatex x' and y' such that:
    x= x'cos(\theta)+ y'sin(\theta)
    y= -x'sin(\theta)+ y' cos(\theta)

    Replace x and y in your equation by those, combine like powers of x' and y' and choose \theta so that the coefficient of x'y' is 0.

    Another method: write the equation in matrix form, \begin{bmatrix}x & y\end{bmatrix}\begin{bmatrix}4 & -\sqrt{2} \\ -\sqrt{2} & 3\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= c.

    Since that is a symmetric matrix, it can be "diagonalized". Find its eigenvalues and eigenvectors. Taking P to be the matrix having the eigenvectors as columns, we can write P^{-1}AP= D where D is the diagonal matrix having the eigenvalues of A on its diagonal. Rewrite the equation x^TAx= c as [tex](x^TP)P{-1}AP(P^{-1}x)= yDy= c[tex]. Since D is diagonal, that equation will have no "xy" term.

    I notice that your "hint" refers to eigenvalues and eigenvectors so apparently you already know the second method. So where is your difficulty? Have you found the eigenvalues? Have your found the eigenvectors?
    Last edited by HallsofIvy; March 22nd 2013 at 05:09 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Mar 2013
    From
    Ljubljana
    Posts
    2

    Re: length of large and small axis of the ellipse

    I did find \lambda_1=5 and \lamda_2=2.

    v_1=(-\sqrt{2}, 1) and v_2=(\frac{1}{\sqrt{2}}, 1).

    Is this correnct and whati is the length of the axis?
    Last edited by mikeno80; March 23rd 2013 at 08:47 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Arrange small squares into one large square?
    Posted in the Geometry Forum
    Replies: 2
    Last Post: February 6th 2013, 03:10 PM
  2. Replies: 2
    Last Post: April 24th 2011, 07:01 AM
  3. Replies: 0
    Last Post: February 23rd 2011, 08:55 AM
  4. Replies: 5
    Last Post: October 18th 2010, 10:38 AM
  5. Graphing very large and very small values
    Posted in the Statistics Forum
    Replies: 1
    Last Post: February 18th 2010, 02:09 PM

Search Tags


/mathhelpforum @mathhelpforum