๐ Topic Summary
Eccentricity is a numerical value that describes how much a conic section (circle, ellipse, parabola, hyperbola) deviates from being a perfect circle. It's denoted by the letter 'e'. A circle has an eccentricity of 0, while other conic sections have eccentricities greater than or equal to 0. Understanding eccentricity helps in classifying and analyzing these important shapes.
The value of eccentricity, $e$, determines the type of conic section:
- โช If $e = 0$, the conic section is a circle.
- ๐ช If $0 < e < 1$, the conic section is an ellipse.
- ๐ช If $e = 1$, the conic section is a parabola.
- hyperbolic geometry โพ๏ธ If $e > 1$, the conic section is a hyperbola.
๐งฎ Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Eccentricity |
A. The set of all points in a plane such that the sum of the distances from two fixed points is constant. |
| 2. Ellipse |
B. A conic section formed when a plane intersects a double cone such that the angle of the plane is greater than the angle of the cone's side. |
| 3. Parabola |
C. A conic section formed when a plane intersects a double cone such that the angle of the plane is equal to the angle of the cone's side. |
| 4. Hyperbola |
D. A measure of how much a conic section deviates from being circular. |
| 5. Circle |
E. The set of all points equidistant from a center point. |
Matching Answers:
- ๐ฏ 1 - D
- ๐ฏ 2 - A
- ๐ฏ 3 - C
- ๐ฏ 4 - B
- ๐ฏ 5 - E
โ๏ธ Part B: Fill in the Blanks
Complete the paragraph with the correct words:
The ______________ of a conic section is a non-negative real number that uniquely characterizes its shape. An __________ has an eccentricity between 0 and 1. A _____________ has an eccentricity of exactly 1. When the eccentricity is 0, the conic section is a ___________. If the eccentricity is greater than 1, we have a ____________.
Possible words: circle, eccentricity, hyperbola, ellipse, parabola
Answers:
- ๐ก eccentricity
- ๐ก ellipse
- ๐ก parabola
- ๐ก circle
- ๐ก hyperbola
๐ค Part C: Critical Thinking
Explain how the eccentricity value affects the shape of an ellipse. Give examples of ellipses with different eccentricity values and describe their shapes.