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Chapter 10: Problem 19

Graph the conics \(r=e /(1-e \cos \theta)\) with \(e=0.4,0.6\) \(0.8,\) and 1.0 on a common screen. How does the value of \(e\) affect the shape of the curve?

Short Answer

Expert verified
As eccentricity \( e \) increases, the ellipse elongates and forms a parabola at \( e = 1 \).

Step by step solution

01

Understand the Equation of a Conic

The conic section is given in polar form as \( r = \frac{e}{1 - e \cos \theta} \). The variable \( e \) represents the eccentricity, which determines the type and shape of the conic.
02

Recall Definitions of Eccentricity

Eccentricity \( e \) defines the shape of the conic: \( e = 0 \) (circle), \( 0 < e < 1 \) (ellipse), \( e = 1 \) (parabola), and \( e > 1 \) (hyperbola). For the provided values, we have ellipses (\( e = 0.4, 0.6, 0.8 \)) and a parabola (\( e = 1.0 \)).
03

Graph Each Value of Eccentricity

Create graphs for each value of \( e \). For \( e = 0.4, 0.6, 0.8 \), plot the ellipses, noting their increasing elongation as \( e \) approaches 1. For \( e = 1.0 \), plot the parabola.
04

Analyze the Effect of Eccentricity

Compare the shapes: as \( e \) increases, the ellipse becomes more elongated. At \( e = 1 \), the curve is no longer closed but becomes a parabola, indicating a transition in conic type.

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Key Concepts

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

Eccentricity
Eccentricity is a critical concept when studying conic sections, as it helps determine the shape and nature of these curves. In general, eccentricity, denoted by the letter \( e \), measures the deviation of a conic section from being circular.
  • For a circle, \( e = 0 \), indicating no deviation; the curve is perfectly circular.
  • For an ellipse, eccentricity falls between 0 and 1 (\( 0 < e < 1 \)), with lower values indicating a shape closer to a circle, and higher values signifying more elongation.
  • When \( e = 1 \), the conic is a parabola, representing another fundamental shape with unique properties.
  • If the eccentricity exceeds 1, the conic is a hyperbola, which opens outward in two separate curves.
Understanding eccentricity helps in predicting the behavior of a curve when graphing it, especially in different coordinate systems like polar coordinates.
Ellipse
An ellipse is one of the primary conic sections defined by an eccentricity between 0 and 1 (\( 0 < e < 1 \)). Its shape resembles an elongated circle. When expressed in polar coordinates, the equation \[ r = \frac{e}{1 - e \cos \theta}\]helps visualize ellipses with various eccentricities. The eccentricity value in the polar equation influences how stretched the ellipse appears. As \( e \) increases, the ellipse elongates along its major axis. Here are some key points about ellipses:
  • All points on an ellipse are the same total distance from two fixed points, known as foci.
  • The center, major axis, and minor axis are important components, with the major axis being the longest diameter.
  • Ellipses appear in many natural and engineered systems, such as planetary orbits and reflectors.
Recognizing and plotting ellipses in polar coordinates provides insight into their geometric properties.
Parabola
The parabola is another essential form of conic sections, emerging when the eccentricity \( e = 1 \). Parabolas are distinctive for their open curve, which extends infinitely in one direction.Parabolas have some fascinating characteristics:
  • Parabolas are defined such that every point is equidistant from a fixed point known as the focus, and a line called the directrix.
  • This property gives parabolas their symmetric shape along an axis known as the axis of symmetry.
  • Parabolas are widely used in various areas, including physics and engineering, particularly when reflecting light or sound, as seen in satellite dishes and flashlight reflectors.
In polar coordinates, when \( e = 1 \), the conic section becomes a parabola, reflecting its open shape which maintains the property of being equidistant from its focus and directrix.
Polar Coordinates
Polar coordinates offer a different way of visualizing and plotting conic sections, transforming traditional Cartesian plots into more circular and radial forms.
  • In polar coordinates, each point on a plane is determined by a distance from a reference point (the origin) and an angle from a reference direction.
  • Polar coordinates are especially useful in scenarios where symmetry concerning a point (rather than a line) is essential.
  • For example, the equation \[ r = \frac{e}{1 - e \cos \theta} \] is used to plot conics in polar form, making it easier to compare different conic sections, like ellipses and parabolas, on the same plot.
Using polar coordinates can greatly simplify the interpretation and analysis of curves, especially when examining eccentricity's effect on the curve's shape.

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Most popular questions from this chapter

Investigate the family of polar curves \(r=1+\cos ^{n} \theta\) where \(n\) is a positive integer. How does the shape change as \(n\) increases? What happens as \(n\) becomes large? Explain the shape for large \(n\) by considering the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates.

Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to determine the parameter interval. $$ \text { One loop of the curve } r=\cos 2 \theta $$

Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity \(\frac{1}{2}, \quad\) directrix \(x=4\)

Investigate the family of curves with polar equations \(r=1+c \cos \theta,\) where \(c\) is a real number. How does the shape change as \(c\) changes?

Find an equation for the conic that satisfies the given conditions. $$ \text { Hyperbola, vertices }(-1,2),(7,2), \quad \text { foci }(-2,2),(8,2) $$
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