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Chapter 10: Problem 31

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=\frac{6}{2+\sin \theta} $$

Short Answer

Expert verified
The curve is an ellipse with eccentricity \( e = 0.5 \).

Step by step solution

01

Identify the General Form

The given polar equation is \( r = \frac{6}{2 + \sin \theta} \). Compare this with the general form \( r = \frac{ed}{1 + e \sin \theta} \) or \( r = \frac{ed}{1 + e \cos \theta} \). Here, \(e\) is the eccentricity and \(d\) is a constant.
02

Match the Equation to the General Form

The given equation, \( r = \frac{6}{2 + \sin \theta} \), can be rewritten to match the standard form by recognizing it as \( r = \frac{6/2}{1 + \frac{1}{2} \sin \theta} \). Thus, \(d = 3\) and \(e = \frac{1}{2}\).
03

Determine the Type of Conic Section

The value of \(e\) determines the type of conic section: if \(0 < e < 1\), the conic is an ellipse; if \(e = 1\), it is a parabola; and if \(e > 1\), it is a hyperbola. Since \(e = \frac{1}{2}\), which is less than 1, the conic is an ellipse.
04

Sketch the Graph

For \( r = \frac{6}{2 + \sin \theta} \), plot points for various angles \( \theta \) from \(0\) to \(2\pi\). Note the symmetry about the horizontal axis and the smaller loop at the far end (towards negative \(x\)-axis). The sketch should show an inner loop, characteristic of a limacon behavior but remains predominantly an ellipse due to \(e < 1\).

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Key Concepts

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

Polar Coordinates
Polar coordinates offer a different way to describe points in a plane compared to the more common Cartesian coordinates (x, y). In polar coordinates, a point's location is defined by two values: its distance from the origin, denoted as \(r\), and the angle, \(\theta\), measured counterclockwise from the positive x-axis. Instead of using a grid-like system, polar coordinates use circles and angles to define positions.
  • The **radius** \(r\) is the distance from the origin to the point.
  • The **angle** \(\theta\) determines the direction of that distance.
This system is particularly useful in problems involving circular or rotational symmetry, making it convenient for dealing with curves like circles, spirals, and certain types of conic sections.
Ellipse
An ellipse is a type of conic section resembling an "oval" shape. It is the set of all points for which the sum of distances to two distinct points, termed the foci, is constant. This property makes ellipses appear frequently in planetary orbits, satellite paths, and lens shapes in optics.
  • If the distance (sum of distances to both foci) is constant, the path traced is an ellipse.
  • An ellipse has two axes: the longer one called the major axis, and the shorter one, the minor axis.
Beyond merely drawing ellipses, we study how their shapes can be described with equations in both Cartesian and polar coordinates. In polar form, the equation highlights the ellipse's center at the origin with eccentricity defining its shape.
Eccentricity
The eccentricity of a conic section is a measure of how much it deviates from being circular. It is denoted by \(e\) and helps in identifying the type of conic section given an equation. The possible values are:
  • If \(0 < e < 1\), the conic is an ellipse.
  • If \(e = 1\), it is a parabola.
  • If \(e > 1\), it is a hyperbola.
In the exercise, we have \(e = \frac{1}{2}\), indicating an ellipse. Greater deviations from \(e = 0\) yield more elongated ellipses. Hence, the eccentricity captures this circular deviation, revealing much about the ellipse's dimensions and properties.
Graph Sketching
Graph sketching involves drawing the curve represented by an equation. It requires understanding the mathematical features assimilated in the equation. In polar coordinates, plotting involves:
  • Identifying key points using various angle values.
  • Noting any symmetry, loops, or deviations particular to the curve.
For the ellipse described by the polar equation \(r=\frac{6}{2+\sin \theta}\), it is vital to focus on symmetry and changes in distance as \(\theta\) varies from \(0\) to \(2\pi\). While plotting, note how \(r\) behaves as \(\theta\) increases, which contributes to the characteristic shape. Additionally, the eccentricity \(e=\frac{1}{2}\) ensures that the curve maintains an elliptical form, despite potential minor loops encountered in polar plots.

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Most popular questions from this chapter

The ends of an elastic string with a knot at \(K(x, y)\) are attached to a fixed point \(A(a, b)\) and a point \(P\) on the rim of a wheel of radius \(r\) centered at \((0,0) .\) As the wheel turns, \(K\) traces a curve \(C\). Find the equation for \(C\). Assume that the string stays taut and stretches uniformly (i.e., \(\alpha=|K P| /|A P|\) is constant).

Graph the curve \(r=\cos (8 \theta / 5)\) using the parametric graphing facility of a graphing calculator or computer. Notice that it is necessary to determine the proper domain for \(\theta\). Assuming that you start at \(\theta=0\), you have to determine the value of \(\theta\) that makes the curve start to repeat itself. Explain why the correct domain is \(0 \leq \theta \leq 10 \pi\)

58\. The path of a projectile fired from level ground with a speed of \(v_{0}\) feet per second at an angle \(\alpha\) with the ground is given by the parametric equations $$ x=\left(v_{0} \cos \alpha\right) t, \quad y=-16 t^{2}+\left(v_{0} \sin \alpha\right) t $$ (a) Show that the path is a parabola. (b) Find the time of flight. (c) Show that the range (horizontal distance traveled) is \(\left(v_{0}^{2} / 32\right) \sin 2 \alpha\) (d) For a given \(v_{0}\), what value of \(\alpha\) gives the largest possible range?

Eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form. Finally, graph the equation showing the rotated axes. 3 x^{2}+10 x y+3 y^{2}+10=0

In order to graph a polar equation such as \(r=f(t)\) using a parametric equation grapher, you must replace this equation by \(x=f(t) \cos t\) and \(y=f(t) \sin t .\) These equations can be obtained by multiplying \(r=f(t)\) by \(\cos t\) and \(\sin t\), respectively. Confirm the discussions of conics in the text by graphing \(r=4 e /(1+e \cos t)\) for \(e=0.1,0.5,0.9,1,1.1\) and \(1.3\) on \([-\pi, \pi]\).
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