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Chapter 12: Problem 41

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
Larger \(e\) results in more elongated ellipses and at \(e=1\), the shape becomes a parabola.

Step by step solution

01

Understand Eccentricity of Conics

The eccentricity \(e\) of a conic section determines its shape. For \(0 < e < 1\), the conic is an ellipse; for \(e = 1\), it is a parabola; and for \(e > 1\), it is a hyperbola. In this problem, all values of \(e\) range from 0.4 to 1.0, so we analyze ellipses and a parabola.
02

Identify Conic Type for Each \(e\)

For \(e = 0.4, 0.6, 0.8\), the conic sections are ellipses because \(0 < e < 1\). When \(e = 1\), the conic becomes a parabola. We do not have \(e > 1\) in this scenario, so no hyperbolas are present.
03

Graph the Polar Equation

Use polar coordinates to graph each given conic equation: \(r = \frac{e}{1-e\cos\theta}\). Plot these equations by calculating \(r\) for a range of \(\theta\), for each \(e\) value, to visualize the ellipse for \(e = 0.4, 0.6, 0.8\), and a parabola for \(e = 1\).
04

Analyze the Effects of \(e\) on Shape

Compare the shapes of the curves: as \(e\) increases from 0.4 to 1.0, the ellipses become less circular and more elongated. At \(e = 1\), the shape is a parabola, which is an open curve compared to a closed ellipse.

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Key Concepts

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

Eccentricity
Eccentricity (\(e\)) is a crucial parameter that helps in distinguishing different types of conic sections. Conics include ellipses, parabolas, and hyperbolas. The eccentricity measures how much a conic section deviates from being circular.
  • When \( 0 < e < 1 \), the conic is an ellipse, and as \( e \) approaches 1, the ellipse becomes more elongated.
  • When \( e = 1 \), the conic turns into a parabola.
  • If \( e > 1 \), the conic is a hyperbola.
Eccentricity provides insight into the shape of the curve, making it a fundamental concept in understanding conics.
Ellipse
An ellipse is a type of conic section where the eccentricity (\(e\)) is less than 1. It is often described as an elongated circle, defined in polar coordinates by the equation \( r = \frac{e}{1-e\cos\theta} \). In this equation, as \(e\) approaches 1, the ellipse becomes flatter and more stretched in one direction.
Here's some additional information about ellipses:
  • Ellipses have two foci, and the sum of the distances from any point on the ellipse to these foci is constant.
  • The larger the eccentricity (closer to 1), the more stretched the ellipse appears.
  • The shape of the ellipse is crucial in many scientific applications, such as planetary orbits and optics.
Ellipses provide important insights into the dynamics of motion and light.
Parabola
A parabola is a unique conic section with an eccentricity (\(e\)) of exactly 1. Unlike ellipses, parabolas have a distinct open shape, meaning they continue infinitely in one direction.
Key features of parabolas include:
  • A single focus and a directrix, each point on the parabola being equidistant from the focus and the directrix.
  • In polar coordinates, it is represented when \( e = 1 \) which takes a unique form because it does not close back on itself.
  • Parabolas are prevalent in real-life applications such as satellite dishes and bridge designs due to their reflective properties.
Understanding parabolas is essential for recognizing these curves in various practical scenarios.
Polar Coordinates
Polar coordinates offer a different perspective compared to traditional Cartesian coordinates. They express points in terms of distance from a reference point (the pole) and an angle from a reference direction. This system is especially beneficial in graphing conic sections like ellipses and parabolas.
Here's why polar coordinates are useful:
  • They simplify the representation of conics centered around a point (the pole).
  • In equations such as \( r = \frac{e}{1-e\cos\theta} \), polar coordinates elegantly describe the curves in terms of \( r \) (radius) and \( \theta \) (angle).
  • Useful for analyzing motion in a circular or orbital motion, as the origin becomes a central point.
Polar coordinates provide clarity and ease in situations where rotational symmetry is present.

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Most popular questions from this chapter

\(29-32\) . (a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$ x^{2}-2 x y+3 y^{2}=8 $$

Find an equation for the hyperbola that satisfies the given conditions. Foci: \(( \pm 6,0),\) vertices: \(( \pm 2,0)\)

\(29-32\) . (a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$ 9 x^{2}-6 x y+y^{2}+6 x-2 y=0 $$

A polar equation of a conic is given. (a) Find the eccentricity and the directrix of the conic. (b) If this conic is rotated about the origin through the given angle , write the resulting equation. (c) Draw graphs of the original conic and the rotated conic on the same screen. $$ r=\frac{9}{2+2 \cos \theta} ; \quad \theta=-\frac{5 \pi}{6} $$

\(23-34\) Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ x^{2}-y^{2}=10(x-y)+1 $$
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