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  1. #1
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    Parabola help

    This is for review and I can't remember this.

    What is the equation of a parabola with focus (6, -10) and directrix x = -2?
    Last edited by Alan; May 21st 2008 at 09:17 AM.
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  2. #2
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    Quote Originally Posted by Alan View Post
    This is for review and I can't remember this.

    What is the equation of a parabola with focus (6, -10) and directrix x = -2?
    Personally I prefer the long way (no surprise to anyone who knows me). A parabola is defined as the locus of points such that the distance between the focus and parabola is the same as the distance between the parabola and the directrix. So
    \sqrt{(x - 6)^2 + (y + 10)^2} = x + 2

    -Dan
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  3. #3
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    The answers I have to choose from are:

    (y+10)^2 = 16(x-2)
    (y+4)^2 = 16(x-6)(y-10)^2 = 16(x+2)
    (y-4)^2 = 16(x+6)

    I'm still kind of confused...
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Alan View Post
    This is for review and I can't remember this.

    What is the equation of a parabola with focus (6, -10) and directrix x = -2?
    See this.

    I prefer the definition I used because I can get equations for parabolas with axes of symmetry that aren't "nice."

    In any event the directrix here is parallel to the y axis, so we can use the form
    (y - k)^2 = \pm 4p(x - h)

    p is the twice the distance between the directrix and the focus, so p = 4. The vertex is half way between the directrix and the focus so it is at V(2, -10). So the equation will be
    (y + 10)^2 = \pm 16(x - 2)

    Finally, we know that the focus is to the right of the directrix, so this parabola opens to the right. Thus we choose the + sign:
    (y + 10)^2 = 16(x - 2)

    Edit:
    You can solve for the answer using my method:
    \sqrt{(x - 6)^2 + (y + 10)^2} = x + 2

    (x - 6)^2 + (y + 10)^2 = (x + 2)^2

    (y + 10)^2 = (x + 2)^2 - (x - 6)^2 = (x^2 + 4x + 4) - (x^2 - 12x + 36)

    (y + 10)^2 = 16x - 32

    (y + 10)^2 = 16(x - 2)

    -Dan
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  5. #5
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    I read the wiki page and tried to understand you but how do you get

    <br />
(y + 10)^2 = (x + 2)^2 - (x - 6)^2 = (x^2 + 4x + 4) - (x^2 - 12x + 36)<br />
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  6. #6
    A riddle wrapped in an enigma
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    Now square both sides



    Transpose (x-6)^2 to the right side of the equation and expand the binomials and simplify.



    The right side simplifies to



    Factor out 16 on the right side

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