Results 1 to 2 of 2

Thread: Eccentricity, trouble w/ the proof

  1. #1
    Newbie
    Joined
    Jul 2011
    Posts
    1

    Eccentricity, trouble w/ the proof

    I'm trying to understand the proof for eccentricity; I can get about halfway through it. The problem I'm having is summarized below

    If e<1, you get the equation of an ellipse of the form

    $\displaystyle \frac{(x-h)^2}{a^2}+\frac{y^2}{b^2}=1$

    where

    Eq. 4:
    $\displaystyle h= -\frac{e^2d}{1-e^2}$
    $\displaystyle a^2= -\frac{e^2d^2}{(1-e^2)^2}$
    $\displaystyle b^2= -\frac{e^2d^2}{1-e^2}$

    The foci of an ellipse are at a distance c from the center, where

    Eq 5:
    $\displaystyle c^2=a^2-b^2=$$\displaystyle \frac{e^4d^2}{(1-e^2)^2}$

    This shows that

    $\displaystyle c= \frac{e^2d}{1-e^2}=-h$

    It follows from equations 4 and 5 that the eccentricity is given by

    $\displaystyle e=\frac{c}{a}$

    If someone could help explain how $\displaystyle e=\frac{c}{a}$ is derived, or really what it means I'd be so grateful. I'm using Calculus Early Transcendentals 6th Ed. by Stewart (Ch. 10, Sect. 6).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Siron's Avatar
    Joined
    Jul 2011
    From
    Belgium
    Posts
    1,254
    Thanks
    24

    Re: Eccentricity, trouble w/ the proof

    The conclusion is that the eccentricity of an ellips is: $\displaystyle e=\frac{c}{a}$, now:
    If you write: $\displaystyle e^2=\frac{c^2}{a^2}$
    Calculating with this:
    $\displaystyle \frac{c^2}{a^2}=\frac{a^2-b^2}{a^2}=1-\frac{b^2}{a^2}$
    Substitution of $\displaystyle b^2$ and $\displaystyle a^2$ out of eq 4 and you get:
    $\displaystyle 1-\frac{b^2}{a^2}=1-\left[\frac{\frac{-e^2d^2}{1-e^2}}{\frac{-e^2d^2}{(1-e^2)^2}}}\right]$
    $\displaystyle =1-(1-e^2)=e^2$
    Now take the square root of $\displaystyle e^2=e$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. having trouble with a proof.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Aug 30th 2010, 12:14 PM
  2. Proof trouble
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Aug 25th 2010, 05:48 PM
  3. Trouble with a proof...help?
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: Apr 28th 2009, 03:45 PM
  4. Trouble with Quick Proof
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: Apr 19th 2009, 12:55 PM
  5. Having trouble formulating proof...
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Nov 26th 2008, 07:56 AM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum