Results 1 to 3 of 3

Math Help - [SOLVED] Equation of an ellipse...

  1. #1
    pidgezero_one
    Guest

    [SOLVED] Equation of an ellipse...

    I'm so sorry if this doesn't belong here. ><; I'm going a bit frantic across homework help forums looking for responses to this. @_@;

    Does anyone know how to calculate the equation of an ellipse (in this case, (x^2/b^2) + (y^2/a^2) = 1 since I graphed the points and it looked like the major axis was vertical) when the only given information is that the center is (0,0) and that 2 points on the ellipse are (2, -2) and (-2/3, 10/3)?

    Any help would be greatly appreciated ^^;

    EDIT Nevermind I solved it. Sorry 'bout that.
    Last edited by pidgezero_one; May 6th 2006 at 04:40 PM. Reason: Solved
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,899
    Thanks
    328
    Awards
    1
    Quote Originally Posted by pidgezero_one
    I'm so sorry if this doesn't belong here. ><; I'm going a bit frantic across homework help forums looking for responses to this. @_@;

    Does anyone know how to calculate the equation of an ellipse (in this case, (x^2/b^2) + (y^2/a^2) = 1 since I graphed the points and it looked like the major axis was vertical) when the only given information is that the center is (0,0) and that 2 points on the ellipse are (2, -2) and (-2/3, 10/3)?

    Any help would be greatly appreciated ^^;

    EDIT Nevermind I solved it. Sorry 'bout that.
    I see that you got it. I'll answer it anyway, for the sake of completeness.

    The general form for an ellipse is:
    \frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1 where the center of the ellipse (the geometric center, not one of the foci) is at the coordinates (h,k).

    Since we know the center is at (0,0) we know that h = k = 0 and the ellipse equation is:
    \frac{x^2}{b^2}+\frac{y^2}{a^2}=1

    Now, we have two points on the ellipse: (2,-2) and (-2/3,10/3). Putting these points into the ellipse equation gives us (respectively)
    1. \frac{4}{b^2}+\frac{4}{a^2}=1
    and
    2. \frac{4}{9b^2}+\frac{100}{9a^2}=1

    So we have two equations for a and b. This should suffice to solve the system. Equation 1 gives us:
    \frac{1}{b^2}=\frac{1}{4}-\frac{1}{a^2}

    Putting this into equation 2 gives us:
    \frac{4}{9} \left ( \frac{1}{4}-\frac{1}{a^2} \right ) +\frac{100}{9a^2}=1

    1 - \frac{4}{a^2} + \frac{100}{a^2}=9

    1+ \frac{96}{a^2}=9

    \frac{a^2}{96}=\frac{1}{8}

    a = \pm \sqrt{\frac{96}{8}}=\pm \sqrt{12}=\pm 2 \sqrt 3

    Since a is the length of the semi-major or semi-minor axis, we choose the positive value.

    This gives a value for b as: b = \sqrt 6.

    Thus the equation for the ellipse is:
    \frac{x^2}{6}+\frac{y^2}{12}=1.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by topsquark
    I see that you got it. I'll answer it anyway, for the sake of completeness.

    The general form for an ellipse is:
    \frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1 where the center of the ellipse (the geometric center, not one of the foci) is at the coordinates (h,k).
    This is the general form of the equation of an ellipse with the major axis
    aligned with either the x-axis or y-axis. Since we are given that the centre is
    at (0,0), we are in general left with three degrees of freedom (eccentricity,
    length of semi-major axis and orientation). Now we are given two points,
    which gives us two equations and three unknowns, so in general there is no
    unique solution.

    So in order to solve this we need the additional information/assumption about the orientation.

    RonL
    Last edited by CaptainBlack; May 7th 2006 at 12:41 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Proof of algebra for ellipse
    Posted in the Algebra Forum
    Replies: 1
    Last Post: April 16th 2010, 02:14 AM
  2. [SOLVED] Ellipse and Circle problem
    Posted in the Pre-Calculus Forum
    Replies: 6
    Last Post: May 11th 2009, 03:37 PM
  3. Ellipse Equation
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: April 28th 2008, 11:36 AM
  4. [SOLVED] Ellipse - Focus and Directrix
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: February 25th 2008, 05:09 AM
  5. [SOLVED] [SOLVED] Find antoher Point on an Ellipse
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 3rd 2007, 12:11 AM

/mathhelpforum @mathhelpforum