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Math Help - [SOLVED] Ellipse - Focus and Directrix

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    [SOLVED] Ellipse - Focus and Directrix

    An ellipse can be determined by means of a focus and a directrix. In the diagram, the directrix of an ellipse is the line x = \frac {25}{3} and a focus is (3,0). The relationship determining the ellipse is PF = \frac {3}{5}PD for all point P.)

    a. Use the distance formula to write and expression for PF, and set it equal to \frac {3}{5}PD. (Hint: You should not need the distance formula for PD.

    b.Simply the equation you wrote in part (a), and put it in the graphing form of an equation of an ellipse. What is the length of the minor axis of this ellipse.

    I posted the picture of the diagram below made in paint as best as it resembles on the paper. Ok...so my question is how do you solve this? I know the equation for and ellipse is \frac {(x-h)^2 }{a^2} + \frac {(y-k)^2}{b^2} = 1. I have know idea though what a directrix is as I haven't learned it yet.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by chrozer View Post
    An ellipse can be determined by means of a focus and a directrix. In the diagram, the directrix of an ellipse is the line x = \frac {25}{3} and a focus is (3,0). The relationship determining the ellipse is PF = \frac {3}{5}PD for all point P.)

    a. Use the distance formula to write and expression for PF, and set it equal to \frac {3}{5}PD. (Hint: You should not need the distance formula for PD.

    b.Simply the equation you wrote in part (a), and put it in the graphing form of an equation of an ellipse. What is the length of the minor axis of this ellipse.

    I posted the picture of the diagram below made in paint as best as it resembles on the paper. Ok...so my question is how do you solve this? I know the equation for and ellipse is \frac {(x-h)^2 }{a^2} + \frac {(y-k)^2}{b^2} = 1. I have know idea though what a directrix is as I haven't learned it yet.
    a) PF = \sqrt{(x - 3)^2 + y^2} = \frac{3}{5} \left ( \frac{25}{3} - x \right )
    (The directrix is simply a line associated with the definition of the ellipse, nothing more.)

    You should be able to work with that to get b) and c). Give it a try and post your attempt if you have trouble.

    -Dan
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    Quote Originally Posted by topsquark View Post
    a) PF = \sqrt{(x - 3)^2 + y^2} = \frac{3}{5} \left ( \frac{25}{3} - x \right )
    (The directrix is simply a line associated with the definition of the ellipse, nothing more.)

    You should be able to work with that to get b) and c). Give it a try and post your attempt if you have trouble.

    -Dan

    Ok I simplified it and got \frac {x^2}{25} + \frac {y^2}{16} = 1, with the minor axis being 4. Is this right? Seems like it's right.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by chrozer View Post
    Ok I simplified it and got \frac {x^2}{25} + \frac {y^2}{16} = 1, with the minor axis being 4. Is this right? Seems like it's right.
    That is correct. Good job!

    -Dan
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    Quote Originally Posted by topsquark View Post
    That is correct. Good job!

    -Dan
    Thnx so much.
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