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Chapter 10: Problem 269

Explain how eccentricity determines which conic section is given.

Short Answer

Expert verified
Eccentricity (\( e \)) classifies conics: \( e = 0 \) (circle), \( 0 < e < 1 \) (ellipse), \( e = 1 \) (parabola), \( e > 1 \) (hyperbola).

Step by step solution

01

Define Eccentricity

Eccentricity is a numerical value that describes the shape of a conic section. It is represented by the letter \( e \). Depending on the value of \( e \), the conic can be an ellipse, parabola, or hyperbola.
02

Determine Circle and Ellipse

For a circle, the eccentricity \( e = 0 \). An ellipse has a value of \( 0 < e < 1 \). This indicates that the closer \( e \) is to 0, the more circular the ellipse becomes.
03

Identify Parabola

A parabola is determined by an eccentricity \( e = 1 \). This is the boundary case between ellipses and hyperbolas.
04

Define Hyperbola

A hyperbola has an eccentricity \( e > 1 \). As \( e \) increases, the branches of the hyperbola become more widely spaced.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conic Sections
Conic sections are the curves that we get by slicing a cone at different angles. Depending on how you cut, you can form different shapes. These shapes include ellipses, parabolas, and hyperbolas. One distinct feature that helps in identifying these shapes is eccentricity, denoted by the letter \( e \). Different values of eccentricity point to different types of conic sections. Understanding these sections is really useful in mathematics and physics, as they often describe the paths of planets and other objects in space.
Ellipse
An ellipse is a type of conic section that resembles an elongated circle. It forms when you cut through a cone at an angle that is not parallel or perpendicular to the cone鈥檚 base. The eccentricity in an ellipse is greater than 0 but less than 1 \((0 < e < 1)\). The closer the value of \( e \) is to 0, the rounder the ellipse becomes, making it very similar to a circle. Ellipses have two foci, and the sum of distances from any point on the ellipse to these foci is constant. You'll often encounter ellipses in astronomical contexts, as they describe the orbits of planets and moons.
Parabola
A parabola is another interesting conic section. It appears when you slice a cone parallel to its side. The eccentricity of a parabola is exactly 1 \((e = 1)\). This makes it a special boundary case between ellipses and hyperbolas. Parabolas have a unique characteristic - a single focus and a directrix, which is a line that helps define the shape. Each point on a parabola is equidistant from the focus and the directrix. This property makes parabolas ideal for reflecting properties, hence their use in satellite dishes and headlights.
Hyperbola
Hyperbolas are formed by cutting a cone with an angle steep enough that the plane intersects both nappes of the cone. The eccentricity of a hyperbola is greater than 1 \((e > 1)\), which means its shape consists of two separate curves, or branches. As \( e \) increases, these branches become more spread out from each other. Hyperbolas have two foci and, like ellipses, have a constant difference in distances from any point on one branch to the two foci. In mathematics, hyperbolas have various applications, including equations that model hyperbolic trajectories and certain types of wave patterns.

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Most popular questions from this chapter

Graph the hyperbola, labeling vertices and foci. $$x^{2}-4 y^{2}+6 x+32 y-91=0$$

Write the equation of the parabola in standard form. Then give the vertex, focus, and directrix. $$(x+2)^{2}=\frac{1}{2}(y-1)$$

Determine which of the conic sections is represented. $$4 x^{2}+14 x y+5 y^{2}+18 x-6 y+30=0$$

Find a polar equation of the conic with focus at the origin, eccentricity of \(e=2,\) and directrix: \(x=3.\)

Identify the conic with focus at the origin, and then give the directrix and eccentricity. $$r=\frac{5}{4+6 \cos \theta}$$
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