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Math Help - Conic sections

  1. #1
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    Conic sections

    5x2 + 6x - 4y = x2 - y2 -2x
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  2. #2
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    Hello, pbeasley!

    No instructions? . . . I'll have to guess . . .


    5x^2 + 6x - 4y \:= \:x^2 - y^2 -2x

    We have: . 4x^2 + 8x + y^2 - 4y \:=\:0 \quad\Rightarrow\quad 4(x^2 + 2x \qquad) + (y^2 - 4y \qquad) \:=\:0<br />

    Complete the square: . 4(x^2 + 2x {\color{blue}\:+\: 1}) + (y^2 - 4y {\color{red}\:+\: 4}) \:=\:0 {\color{blue}\:+\:4} {\color{red}\:+\: 4}

    And we have: . 4(x+1)^2 + (y-2)^2 \:=\:8 \quad\Rightarrow\quad \frac{(x+2)^2}{2} + \frac{(y-2)^2}{8} \:=\:1


    We have an ellipse.

    . . Center: (-1,2)
    . . Major axis (vertical): a = 2\sqrt{2}
    . . Minor axis (horizontal):  b = \sqrt{2}
    . . Focal distance: c = \sqrt{6}
    . . Eccentricity: e = \tfrac{\sqrt{3}}{2}

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