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Chapter 11: Problem 74

Graph the lines and conic sections in Exercises \(65-74.\) $$r=1 /(1+2 \cos \theta)$$

Short Answer

Expert verified
The graph is a hyperbola in polar coordinates.

Step by step solution

01

Identify the type of conic section

The given polar equation is \(r = \frac{1}{1 + 2 \cos \theta}\). This equation is in the standard form of a conic section in polar coordinates, \(r = \frac{ed}{1 + e \cos \theta}\), where e is the eccentricity. Here, \(e = 2\), which means that this conic is a hyperbola because \(e > 1\).
02

Determine parameters

Since we have identified the conic as a hyperbola with \(e = 2\), let's analyze the rest of the equation. Here, \(d = 1\) represents the semi-latus rectum. The equation \(r = \frac{ed}{1 + e \cos \theta}\) shows that the conic is centered such that its directrix is perpendicular to the initial line (polar axis).
03

Graph the polar equation

To graph the polar equation, we can start by plotting points for various values of \(\theta\) and compute \(r\). For example, for \(\theta = 0\), \(r = \frac{1}{1+2} = \frac{1}{3}\). For \(\theta = \frac{\pi}{2}\), \(r = \frac{1}{1} = 1\). Continue this for other key angles such as \(\pi\), \(\frac{3\pi}{2}\), etc., to get symmetric behavior. These values will define the general shape of the hyperbola in polar form.

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Key Concepts

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

Hyperbola
A hyperbola is a type of conic section. It appears as two separate curves facing away from each other. This happens when slicing through a cone with a plane that is steeper than the cone's side. In polar coordinates, the standard form of a hyperbola is written as \( r = \frac{ed}{1 + e \cos \theta} \), where \( e \) is the eccentricity and \( d \) is the semi-latus rectum. The hyperbola's defining property is that its eccentricity \( e\) is greater than 1.

The parts of a hyperbola include:
  • *Vertices*: points where each branch makes its closest approach to the other branch.
  • *Foci*: two fixed points inside each branch, used to form the hyperbola.
  • *Equations*: derived from the distance properties related to the foci.
Hyperbolas in polar coordinates often appear when the angle \( \theta \) changes as you graph points. As you plot more points, you'll see the hyperbola's distinct shape.
Eccentricity
Eccentricity is a crucial parameter in determining the shape of conic sections, including hyperbolas. It is denoted by the letter \( e \). Eccentricity measures how much a conic section deviates from being circular. For hyperbolas, the eccentricity is always greater than 1. The greater the eccentricity, the more "stretched" or elongated the hyperbola appears.

Here are key points about eccentricity:
  • For a circle, \( e = 0 \). This means circles have zero deviation from circularity.
  • For an ellipse, \( 0 < e < 1 \).
  • For a parabola, \( e = 1 \).
  • For a hyperbola, \( e > 1 \).
In our specific example, the eccentricity \( e = 2 \). This firmly classifies the curve as a hyperbola. Eccentricity influences how we graph and interpret these shapes in different coordinate systems. Understanding \( e \) helps identify the nature of the curve easily.
Conic Sections
Conic sections refer to the shapes created when a plane intersects a double-napped cone. These shapes are circles, ellipses, parabolas, and hyperbolas. Each shape has unique properties and equations to describe them. Understanding conic sections is essential in various fields such as astronomy, physics, and engineering.

Several features set conic sections apart:
  • *Circle*: Formed when the plane is perpendicular to the cone's axis. It has eccentricity \( e = 0 \).
  • *Ellipse*: Occurs when the plane's angle is less than the cone's side but not perpendicular, having \( 0 < e < 1 \).
  • *Parabola*: The plane is parallel to the cone's side, leading to \( e = 1 \).
  • *Hyperbola*: The plane is steeper than the cone's sides, with \( e > 1 \).
Conic sections can also be expressed in both Cartesian and polar coordinates. Polar coordinates are particularly useful in representing these shapes because they allow for easier computation and understanding of orbits and other natural phenomena.

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Most popular questions from this chapter

Replace the Cartesian equations in Exercises \(53-66\) with equivalent polar equations. $$x^{2}+y^{2}=4$$

Find the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections in Exercises \(57-68 .\) $$ x^{2}+2 y^{2}-2 x-4 y=-1 $$

The witch of Maria Agnesi The bell-shaped witch of Maria Agnesi can be constructed in the following way. Start with a circle of radius \(1,\) centered at the point \((0,1),\) as shown in the accompanying figure. Choose a point \(A\) on the line \(y=2\) and connect it to the origin with a line segment. Call the point where the segment crosses the circle \(B\) . Let \(P\) be the point where the vertical line through \(A\) crosses the horizontal line through \(B\) . The witch is the curve traced by \(P\) as \(A\) moves along the line \(y=2\) . Find parametric equations and a parameter interval for the witch by expressing the coordinates of \(P\) in terms of \(t\) the radian measure of the angle that segment \(O A\) makes with the positive \(x\) -axis. The following equalities (which you may assume) will help. $$\begin{array}{ll}{\text { a. } x=A Q} & {\text { b. } y=2-A B \sin t} \\\ {\text { c. } A B \cdot O A=(A Q)^{2}}\end{array}$$

Find the point on the parabola \(x=t, y=t^{2},-\infty< t <\infty\) closest to the point \((2,1 / 2) .\) (Hint: Minimize the square of the distance as a function of \(t\). )

Exercises \(53-56\) give equations for hyperbolas and tell how many units up or down and to the right or left each hyperbola is to be shifted. Find an equation for the new hyperbola, and find the new center, foci, vertices, and asymptotes. $$ \frac{y^{2}}{3}-x^{2}=1, \quad \text { right } 1, \text { up } 3 $$
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