# Conics.

• May 8th 2010, 04:03 PM
Anemori
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?
• May 12th 2010, 10:19 AM
Soroban
Hello, Anemori!

Quote:

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

Quote:

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