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Math Help - Ellipse centered at a focus

  1. #1
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    Ellipse centered at a focus

    I want to get the polar equation for a ellipse in polar form with the center at one of the foci. I started with the equation of an ellipse in Cartesian coordinates:
    \frac{x^2}{a^2}+\frac{y^2}{b^2}=1
    Using x=rcos\theta and  y=rsin\theta I obtained:
    \frac{r^2cos^2\theta}{a^2}+\frac{r^2sin^2\theta}{b  ^2}=1
    b^2r^2cos^2\theta+a^2r^2sin^2\theta=a^2b^2
    r^2(b^2cos^2\theta+a^2sin^2\theta)=a^2b^2
    r=\frac{ab}{\sqrt{(b^2cos^2\theta+a^2sin^2\theta)}  }

    Now I would like the same thing but centered at a focus:
    From wiki, I found that the distance between the foci is \sqrt{a^2-b^2}. I not sure why that is the case and it would be nice to see some proof. Then I reasoned the ellipse must be shifted half that distance as below:
    \frac{(x-\frac{\sqrt{a^2-b^2}}{2})^2}{a^2}+\frac{y^2}{b^2}=1
    \frac{x^2+\frac{a^2-b^2}{4}-\sqrt{a^2-b^2}}{a^2}+\frac{y^2}{b^2}=1
    \frac{x^2}{a^2}+\frac{1}{4}(1-\frac{b^2}{a^2})-\frac{1}{a}\sqrt{1-\frac{b^2}{a^2}}+\frac{y^2}{b^2}=1
    Using the identity e=\sqrt{1-\frac{b^2}{a^2}} I obtained:
    \frac{x^2}{a^2}+\frac{e^2}{4}-\frac{e}{a}+\frac{y^2}{b^2}=1
    \frac{r^2cos^2\theta}{a^2}+\frac{e^2}{4}-\frac{e}{a}+\frac{r^2sin^2\theta}{b^2}=1
    \frac{r^2cos^2\theta}{a^2}+\frac{r^2sin^2\theta}{b  ^2}=\frac{e}{a}-\frac{e^2}{4}
    r^2\frac{a^2sin^2\theta+b^2cos^2\theta}{a^2b^2}=\f  rac{e(4-ae)}{4a}
    r^2=\frac{e(4-ae)}{4a}\frac{a^2b^2}{a^2sin^2\theta+b^2cos^2\thet  a}
    r^2=\frac{e(4-ae)}{4}\frac{ab^2}{a^2(1-cos^2\theta)+b^2cos^2\theta}
    r^2=\frac{e(4-ae)}{4}\frac{ab^2}{a^2-(a^2-b^2)cos^2\theta}
    r^2=\frac{e(4-ae)}{4}\frac{\frac{b^2}{a}}{1-\frac{a^2-b^2}{a^2}cos^2\theta}
    r^2=\frac{e(4-ae)}{4}\frac{\frac{b^2}{a}}{1-e^2cos^2\theta}

    I'm kindda stuck here. I know that I should get something like r=\frac{a(1-e^2)}{1+ecos\theta}
    Therefore:
    r^2=\frac{a^2(1-2e^2+e^4)}{1+e^2cos^2\theta+2ecos\theta}
    Where will the cos\theta term came from? and the e^4? Am I on the right track?
    Am I doing something wrong?
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  2. #2
    MHF Contributor Amer's Avatar
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    for ellipse use this sub

    \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

    let \frac{x}{a} = r\cos \theta , \frac{y}{b} = r\sin \theta so

    r^2 \cos ^2 \theta + r^2 \sin ^2 \theta =1

    r^2 = 1 and do same sub for the second question
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  3. #3
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    By the way, you do NOT want to get the "center" at one of the foci- that is impossible. You want to have the origin of the coordinate system at one focus.
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    By the way, you do NOT want to get the "center" at one of the foci- that is impossible. You want to have the origin of the coordinate system at one focus.
    \lim_{\text{foci} \rightarrow \text{centre}} \text{Ellipse} = \text{Circle}.
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  5. #5
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    Quote Originally Posted by Amer View Post
    for ellipse use this sub

    \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

    let \frac{x}{a} = r\cos \theta , \frac{y}{b} = r\sin \theta so

    r^2 \cos ^2 \theta + r^2 \sin ^2 \theta =1

    r^2 = 1 and do same sub for the second question
    No I can't do that. That's just a circle with radius = 1. What I want to do is have the equation of an ellipse with the center of the coordinate system centered at one of the foci. From wiki, I know that the answer should be:
    r=\frac{a(1-e^2)}{1+ecos\theta} with e=\sqrt{1-\frac{a^2}{b^2}}

    I am actually using this in a physics problem showing that the orbital velocity at the point which the minor axis meets the elliptical orbit is the geometrical mean of the velocities at the ends of the major axis. With this identity: r=\frac{a(1-e^2)}{1+ecos\theta}
    my proof works very well, but I don't want to just pull an equation off wiki and rest my proof on that. I would like to derive that equation by doing a shift of coordinates.
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  6. #6
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    Quote Originally Posted by synclastica_86 View Post
    No I can't do that. That's just a circle with radius = 1. What I want to do is have the equation of an ellipse with the center of the coordinate system centered at one of the foci. From wiki, I know that the answer should be:
    r=\frac{a(1-e^2)}{1+ecos\theta} with e=\sqrt{1-\frac{a^2}{b^2}}

    I am actually using this in a physics problem showing that the orbital velocity at the point which the minor axis meets the elliptical orbit is the geometrical mean of the velocities at the ends of the major axis. With this identity: r=\frac{a(1-e^2)}{1+ecos\theta}
    my proof works very well, but I don't want to just pull an equation off wiki and rest my proof on that. I would like to derive that equation by doing a shift of coordinates.
    The eccentricity of a circle is equal to 1. A circle is an ellipse whose foci are at the centre.
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