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Math Help - Is my work correct?

  1. #1
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    Is my work correct?

    Question 1: Find the vertex, focus, and directrix of the parabola:
    4x - y^2 - 2y - 33 = 0

    My work:
    y^2 + 2y = 4x - 33
    (y + 1)^2 = 4(x - 8)
    Vertex: (8, -1); Focus: (-7, -1); Directrix: x = -9


    Question 2: Find the vertex, focus, and directrix of the parabola:
    x^2 - 2x + 8y + 9 = 0

    My work:
    8y = -(x^2 - 2x + 9)
    8y = -(x - 1)^2 - 8
    8(y + 1) = -(x - 1)^2
    Vertex: (1, -1); Focus: (0, -1); Directrix: x = -2

    I am not sure if my work is correct, especially in the 2nd question. Can you please tell me if I went wrong?
    Thanks!
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  2. #2
    A riddle wrapped in an enigma
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    Quote Originally Posted by iluvmathbutitshard View Post
    Question 1: Find the vertex, focus, and directrix of the parabola:
    4x - y^2 - 2y - 33 = 0

    My work:
    y^2 + 2y = 4x - 33
    (y + 1)^2 = 4(x - 8)
    Vertex: (8, -1); Focus: (-7, -1); Directrix: x = -9

    Hi iluvmathbutitshard,

    I only have time for the first one. Your vertex looks ok, but I'm getting something different for the focus and directrix.

    4x-y^2-2y-33=0

    4x=y^2+2y+33

    4x=(y^2+2y+1)+33-1

    4x=(y+1)^2+32

    x=\frac{1}{4}(y+1)^2+8

    Vertex (h, k)=(8, -1)

    Focus \left(h+\frac{1}{4a}, k\right)

    This would make the focus \left(8+\frac{1}{4(\frac{1}{4})}, -1\right)=(9, -1)

    And the directrix x=h-\frac{1}{4a}\Longrightarrow x=8-\frac{1}{4(\frac{1}{4})} \Longrightarrow 8-1 = 7
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  3. #3
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    For the second question, the vertex is correct but the focus and directrix are incorrect. The focus should be two units below the vertex at (1, -3) and the directrix is y = 1.
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  4. #4
    A riddle wrapped in an enigma
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    Quote Originally Posted by iluvmathbutitshard View Post

    Question 2: Find the vertex, focus, and directrix of the parabola:
    x^2 - 2x + 8y + 9 = 0

    My work:
    8y = -(x^2 - 2x + 9)
    8y = -(x - 1)^2 - 8
    8(y + 1) = -(x - 1)^2
    Vertex: (1, -1); Focus: (0, -1); Directrix: x = -2

    I am not sure if my work is correct, especially in the 2nd question. Can you please tell me if I went wrong?
    Thanks!
    Your vertex looks good to me, but take another look at your Focus and Directrix. Here's my approach.

    x^2-2x+8y+9=0

    8y=-x^2+2x-9

    8y=-1(x^2-2x+1)-9+1

    y=-\frac{1}{8}(x-1)^2-1

    Vertex (1, -1)

    Focus \left(h, k+\frac{1}{4a}\right)

    Directrix y=k-\frac{1}{4a}
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