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Math Help - Ellipse algebra and transformation

  1. #1
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    Ellipse algebra and transformation

    I am finding it difficult the following problem:

    I have the ellipse (3x^2 + 5y^2 = 75) and I have solved it as follows:

    a=5
    b=sqrt(15)
    Foci: + and - (sqrt(10), 0)
    Directrices: + and - 5*sqrt(10)/2
    Eccentricity: sqrt(10)/5
    Center: (0,0)
    Vertices: (5,0) (-5,0) (0, sqrt(15)), (0, -sqrt(15))

    Now I must equate PF=ePd to the above information. I must show all the workings to "show that the equation holds where the conic intersects with the x-axis". The difficulty is placing all this square roots to fit the algebra. They prefer it that we put all of this in square roots, not decimals. Pity!

    Then I must describe the following translation in terms of the above ellipse: 3x^2 + 12x + 5y^2 - 10y -58 = 0 . I must solve the algebra to find exact coordinates of the points for center, vertices, axes of symmetry and and slopes of any asymptotes.

    I tried to solve this for the center and found the answer (-6,5) but it does not look right so I can not continue.

    Finally, must I write parametric equations for the two ellipses.

    Many thanks for some help is someone can!

    Brigitte
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  2. #2
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    Quote Originally Posted by dolkam View Post
    I am finding it difficult the following problem:

    ...
    Then I must describe the following translation in terms of the above ellipse: 3x^2 + 12x + 5y^2 - 10y -58 = 0 .

    ...
    1. Complete the squares:

    3x^2 + 12x + 5y^2 - 10y -58 = 0~\implies~3(x^2 + 4x\bold{\color{red}+4}) + 5(y^2 - 2y \bold{\color{blue}+1})=58\bold{\color{red}+12} \bold{\color{blue}+5}

    2. You'll get:

    3(x+2)^2+5(y-1)^2=75

    Now continue!
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  3. #3
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    So, please tell me if this is correct:

    (x+2)^2/(75/3) + (y-1)^2/75/5 = 1 ... etc

    Does this make that the transformation is to the new ellipse center (-2, 1)? So that will be 2 units to the left and 1 unit to the top?

    Thank you so much on this part. Maybe I can progress.

    Brigitte
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  4. #4
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    Quote Originally Posted by dolkam View Post
    So, please tell me if this is correct:

    (x+2)^2/(75/3) + (y-1)^2/75/5 = 1 ... etc

    Does this make that the transformation is to the new ellipse center (-2, 1)? So that will be 2 units to the left and 1 unit to the top?

    Thank you so much on this part. Maybe I can progress.

    Brigitte
    Correct!
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  5. #5
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    Ok, please tell me if these following are correct:

    vertices: (3,0), (-7,1), (0, 1+sqrt(15)), (0, 1+(-sqrt(15))
    axes of symmetry: (does this means the minor and major axis of symmetry? if yes, how do I mention the exact coordinates as the axis exend from the points of the vertices mentioned above.)
    slopes of any asymptotes: is there none in this example, since they are only on the hyperbolas?

    Thank you again!
    Brigitte
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  6. #6
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    Quote Originally Posted by dolkam View Post
    Ok, please tell me if these following are correct:

    vertices: (3,1), (-7,1), (0, 1+sqrt(15)), (0, 1+(-sqrt(15))<<<<<< typo

    axes of symmetry: (does this means the minor and major axis of symmetry? if yes, how do I mention the exact coordinates as the axis exend from the points of the vertices mentioned above.)
    ...
    The axes of symmetry are placed on straight lines whose equation you know. Additionally you can state the length of the axes. If I were you I would label the vertices and then you can describe the axes by their endpoints.

    For instance: V_L(3,1) \text{  and  } V_R(-7,1) then the major axis is determined by 2a = \overline{V_L V_R}
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