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Math Help - Ellipse Proof

  1. #1
    Junior Member lanierms's Avatar
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    Ellipse Proof

    Ellipse Proof-ellipse.png Use the following ellipse : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (F, F' are foci of the ellipse and the line passes through it.)

    Prove : \frac{1}{PF}+\frac{1}{QF} = \frac{2a}{b^2}

    I tried to prove it using directrix, but it got way too complicated.
    Last edited by lanierms; September 8th 2012 at 06:33 AM.
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  2. #2
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Ellipse Proof

    Quote Originally Posted by lanierms View Post
    Click image for larger version. 

Name:	Ellipse.PNG 
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ID:	24734 Use the following ellipse : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (F, F' are foci of the ellipse and the line passes through it.)

    Prove : \frac{1}{PF}+\frac{1}{QF} = \frac{2a}{b^2}

    I tried to prove it using directrix, but it got way too complicated.
    If the equality is true for all points P & Q on ellipse then letting P(a,0) & Q(-a,0) we will have:

    e\text{:=}\sqrt{1-\frac{b^2}{a^2}}

    \frac{1}{\text{PF}}+\frac{1}{\text{QF}} =\frac{1}{a-ea}+\frac{1}{a+ea}= \frac{2 a}{b^2}

    In the general case when P & Q are elsewhere on the ellipse. We have:

    Move x-y coordinate's origin to focus F...then PF & QF will be:

    \text{PF}\text{:=}a(1-e \text{Cos}[\theta ])
    \text{QF}\text{:=}a(1-e \text{Cos}[\theta -\pi ])

    \frac{1}{\text{PF}}+\frac{1}{\text{QF}} = \frac{2}{a-a e^2 \text{Cos}[\theta ]^2} = -\frac{2}{-a+\left(a-\frac{b^2}{a}\right) \text{Cos}[\theta ]^2}
    Last edited by MaxJasper; September 8th 2012 at 08:34 PM.
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  3. #3
    Junior Member lanierms's Avatar
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    Re: Ellipse Proof

    It seems like a nice proof, but can there be another way without using eccentricity?
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  4. #4
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Ellipse Proof

    Variation of \frac{1}{\text{PF}}+\frac{1}{\text{QF}} by \theta fpr a=3, b=2

    Attached Thumbnails Attached Thumbnails Ellipse Proof-ellipse-proof2.png  
    Last edited by MaxJasper; September 9th 2012 at 04:30 PM.
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  5. #5
    Junior Member lanierms's Avatar
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    Re: Ellipse Proof

    I don't quite understand the part where you said
    PF = a(1-ecos(theta))

    Can you please explain more about it?
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  6. #6
    Senior Member MaxJasper's Avatar
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    Re: Ellipse Proof

    OF = e*a
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