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Math Help - Focus, directrix & vertex of parabola

  1. #1
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    Exclamation Focus, directrix & vertex of parabola

    Hi

    Find the focus, directrix and vertex of the parabola x = y^2 - 6y + 3.

    Are the answers:

    Focus ( -5\frac{3}{4}, 3) OR (-5, 3)

    Directrix x = - 6 \frac{1}{4} OR x = - 7

    Vertex (-6, 3)

    If the answers are the former, could someone please show me how to do this question?

    Thanx

    \sqrt{\frac{d^2}{n}}
    Last edited by xwrathbringerx; January 17th 2009 at 03:42 PM.
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  2. #2
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    Quote Originally Posted by xwrathbringerx View Post
    Hi

    Find the focus, directrix and vertex of the parabola x = y^2 - 6y + 3.

    Are the answers:

    Focus ( -5\frac{3}{4}, 3) OR (-5, 3)

    Directrix x = - 6 \frac{1}{4} OR x = - 7

    Vertex (-6, 3)

    If the answers are the former, could someone please show me how to do this question?

    Thanx
    Use the formulae here Parabola - Wikipedia, the free encyclopedia but interchange x and y ....
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  3. #3
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    [IMG]file:///C:/DOCUME%7E1/HP_ADM%7E1/LOCALS%7E1/Temp/moz-screenshot.jpg[/IMG][IMG]file:///C:/DOCUME%7E1/HP_ADM%7E1/LOCALS%7E1/Temp/moz-screenshot-1.jpg[/IMG]


    your answers are correct...


    the given equation is x=y^2-6y+3
    it can be written as
    x=(y-3)^2 -6
    or x+6=(y-3)^2
    now shift the coordinate system such that x+6=X and y-3=Y
    then the equation becomes Y^2=X
    compare it with Y^2=4aX
    we get a=1/4

    Vertex X=o,Y=o but in new coordinate system i.e. XY system
    or x+6=0 ,y-3=0 in orig. coordinate system i.e. xy system
    i.e. (-6,3)

    now

    Focus=(1/4,0) but in new coordinate system i.e.XY system
    or X=1/4
    or x+6=1/4
    or x=1/4-6= -23/4

    similarly Y=0 gives y=3
    so the focus is (, 3)

    we know that the equation of diretrix is X=-a
    so, diretrix is
    X=-1/4 but in new coordinate system i.e. XY system

    or x+6=-1/4 in orig. system i.e. xy system

    or x=-25/4

    .
    Last edited by sumit2009; January 17th 2009 at 07:58 PM. Reason: typing error
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