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Math Help - Conics.

  1. #1
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    Conics.

    Hello!

    I have like 3 problems:

    a. Produce a conic system of inequalities representing the "doughnut-shape" created by an ellipse inside a circle.

    b. Produce a conic system that has exactly two points of intersection and consists of a circle and ellipse with the same area.

    c. True or False? Every system of two conic equations whose xy-term is nonexistent has at most 4points of intersection.

    I can start with a circle... I don't know how to start the equation for ellipse and what does it look like iside the circle?
    Last edited by mr fantastic; May 8th 2010 at 03:09 PM. Reason: Re-titled.
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  2. #2
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    Hello, Anemori!

    a. Produce a conic system of inequalities representing the "doughnut-shape"
    created by an ellipse inside a circle.
    Code:
                    |
                ..* * *..
              *:::::|:::::*
            *:::::::|:::::::*
           *::::::::|::::::::*
           :::::::::**::::::::
          *::::*    |b   *::::* c
      - - *:-:* - - + - - *:-:* - - -
          *::::*    |  a *::::*
            :::::::***::::::::
           *::::::::|::::::::*
            *:::::::|:::::::*
              *:::::|:::::*
                  * * *
                    |

    The equation of the ellipse is: . \frac{x^2}{a^2} + \frac{y^2}{b^2} \:=\:1

    The equation of the circle is: . x^2 + y^2 \:=\:c^2,\,\text{ where }c \,\geq\,a,b


    System: . \begin{Bmatrix} \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} & \geq & 1 \\ \\[-3mm]x^2 + y^2 & \leq & c^2 \end{Bmatrix}




    b. Produce a conic system that has exactly two points of intersection
    and consists of a circle and ellipse with the same area.

    The equation of an ellipse is: . \frac{x^2}{a^2} + \frac{y^2}{b^2} \:=\:1
    . . Its area is: . \pi ab

    The equation of a circle is: . x^2 + y^2 \:=\:r^2
    . . Its area is: . \pi r^2

    The areas are equal: . \pi ab \:=\:\pi r^2 \quad\Rightarrow\quad ab \:=\:r^2

    Let: . a = 4,\;b = 1,\;r = 2


    The ellipse is: . \frac{x^2}{16} + \frac{y^2}{1} \;=\;1
    Code:
                   1|
                  o o o
            o       |       o
         o          |          o
      - o - - - - - + - - - - - o - -
         o          |          o 4
            o       |       o
                  o o o
                    |


    The circle is: . x^2 + y^2 \:=\:4
    Code:
                   2|
                  ◊ ◊ ◊
              ◊     |     ◊
             ◊      |      ◊
                    |
        - - ◊ - - - + - - - ◊ - -
                    |        2
             ◊      |      ◊
              ◊     |     ◊
                  ◊ ◊ ◊
                    |



    Move the circle 2 units to the right.

    Code:
                    |       :
                    |     ◊ ◊ ◊
                  o o ♥     :     ◊
            o       |◊      o      ◊
         o          |       :  o
    - - o - - - - - ◊ - - - + - o - ◊ - - 
         o          |       :  o
            o       |◊      o      ◊
                  o o ♥     :
                    |     ◊ ◊ ◊

    System: . \begin{Bmatrix}\dfrac{x^2}{16} + \dfrac{y^2}{1} \:=\:1 \\ \\[-3mm] (x-2)^2 + y^2 \:=\:4 \end{Bmatrix}

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