# Conics.

#### Anemori

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

MHF Hall of Honor
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: .$$\displaystyle \begin{Bmatrix}\dfrac{x^2}{16} + \dfrac{y^2}{1} \:=\:1 \\ \\[-3mm] (x-2)^2 + y^2 \:=\:4 \end{Bmatrix}$$

• Anemori
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