Conics.

Jan 2010
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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?
 
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Soroban

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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: .\(\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} \:=\:1\)

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


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




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: .\(\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} \:=\:1\)
. . Its area is: .\(\displaystyle \pi ab\)

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

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

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


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


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

 
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