Hello, Anemori!
a. Produce a conic system of inequalities representing the "doughnutshape"
created by an ellipse inside a circle. Code:

..* * *..
*::::::::::*
*::::::::::::::*
*::::::::::::::::*
:::::::::**::::::::
*::::* b *::::* c
  *::*   +   *::*   
*::::*  a *::::*
:::::::***::::::::
*::::::::::::::::*
*::::::::::::::*
*::::::::::*
* * *

The equation of the ellipse is: .
The equation of the circle is: .
System: .
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: .
. . Its area is: .
The equation of a circle is: .
. . Its area is: .
The areas are equal: .
Let: .
The ellipse is: . Code:
1
o o o
o  o
o  o
 o      +      o  
o  o 4
o  o
o o o

The circle is: . 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: .