📚 Topic Summary
The eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. It tells us whether the conic section is a circle, ellipse, parabola, or hyperbola. A circle has an eccentricity of 0, an ellipse has an eccentricity between 0 and 1, a parabola has an eccentricity of 1, and a hyperbola has an eccentricity greater than 1. Understanding eccentricity helps predict the behavior and appearance of these curves in various applications, from optics to astronomy. Think of it as a measure of how 'un-circular' a conic section is! 📏
Eccentricity ($e$) is calculated differently for each type of conic section:
- 🔵 Circle: $e = 0$
- 🥚 Ellipse: $e = \frac{c}{a}$, where $c$ is the distance from the center to a focus and $a$ is the semi-major axis.
- 🔪 Parabola: $e = 1$
- ♾️ Hyperbola: $e = \frac{c}{a}$, where $c$ is the distance from the center to a focus and $a$ is the semi-transverse axis.
🔤 Part A: Vocabulary
Match the term with its correct definition.
| Term |
Definition |
| 1. Focus |
A. A conic section where the eccentricity is equal to 1. |
| 2. Eccentricity |
B. A conic section where the eccentricity is between 0 and 1. |
| 3. Parabola |
C. A point used in defining a conic section. |
| 4. Ellipse |
D. The ratio determining the shape of a conic section. |
| 5. Hyperbola |
E. A conic section where the eccentricity is greater than 1. |
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms.
The ________ of a conic section determines its shape. A circle has an eccentricity of ________. An ________ has an eccentricity between 0 and 1. A ________ has an eccentricity of 1, and a ________ has an eccentricity greater than 1.
🤔 Part C: Critical Thinking
Explain how the eccentricity of an ellipse changes as it approaches a circle, and what happens as it approaches a line segment.