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Chapter 11: Problem 12

How does the eccentricity determine the type of conic section?

Short Answer

Expert verified
Answer: The eccentricity (e) is a value that determines the shape of a conic section, and it is connected to the foci of the conic. It affects the shape of each type of conic section as follows: 1. e = 0: Circle - The foci coincide at the center, and the distance from any point on the circle to the center is constant. 2. 0 < e < 1: Ellipse - The ellipse's shape becomes more elongated as the eccentricity increases, with the foci approaching the ends of the major axis and a constant total distance from any point on the ellipse to both focal points. 3. e = 1: Parabola - The conic section becomes unbounded, with only one focus and equal distance from any point on the parabolic curve to the focus and a line called the directrix. 4. e > 1: Hyperbola - The conic section becomes two distinct unbounded branches, with two foci and a relationship involving the difference of distances from each focus to points on the two branches, and the shape becomes more elongated as the eccentricity increases.

Step by step solution

01

Define Eccentricity

Eccentricity (e) is a parameter that determines the shape of a conic section. It is a non-negative value and is connected to the foci of the conic. The foci are crucial points that affect the shape of the conic section, and the distance from any point on the conic to these foci is used to describe the conic.
02

Eccentricity and Circles

A circle has an eccentricity of 0 (e=0). In this case, the two foci coincide at the center of the circle, and the distance from any point on the circle to the center is constant. A circle can be thought of as a "non-stretched" ellipse since it is perfectly symmetrical and the foci are in the same location.
03

Eccentricity and Ellipses

An ellipse has an eccentricity between 0 and 1 (0<e<1). As the eccentricity increases from 0, the ellipse's shape becomes more elongated while maintaining the same center. The two foci approach the ends of the major axis, and the total distance from any point on the ellipse to both focal points remains constant.
04

Eccentricity and Parabolas

A parabola has an eccentricity equal to 1 (e=1). When the eccentricity reaches 1, the conic section becomes unbounded, and the shape is no longer an enclosed curve. In the case of a parabola, there is only one focus, and the distance from any point on the parabolic curve to the focus is equal to the distance from that point to a line called the directrix.
05

Eccentricity and Hyperbolas

A hyperbola has an eccentricity greater than 1 (e>1). When the eccentricity exceeds 1, the conic section becomes two distinct unbounded curves called branches. In this case, there are still two foci, but the mathematical relationship that describes the conic section now involves the difference of the distances from each focus to points on the two branches. Just like in ellipses, as the eccentricity increases, the shape of the hyperbola becomes more elongated. In conclusion, the eccentricity determines the type of conic section as follows: 1. e = 0: Circle 2. 0 < e < 1: Ellipse 3. e = 1: Parabola 4. e > 1: Hyperbola

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