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Chapter 10: Problem 5

Consider the polar equation \(r=\frac{4}{1+e \sin \theta}\) (a) Use a graphing utility to graph the equation for \(e=0.1\), \(e=0.25, e=0.5, e=0.75\), and \(e=0.9 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{-}\) and \(e \rightarrow 0^{+}\) (b) Use a graphing utility to graph the equation for \(e=1\). Identify the conic. (c) Use a graphing utility to graph the equation for \(e=1.1\), \(e=1.5\), and \(e=2 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{+}\) and \(e \rightarrow \infty\)

Short Answer

Expert verified
The polar equation, for values \(0 \leq e < 1\), represents circles and ellipses transitioning to a parabola as \(e\rightarrow 1^{-}\). For \(e=1\), it represents a parabola. For values \(e > 1\), it represents hyperbolas becoming more linear as \(e \rightarrow \infty\).

Step by step solution

01

Plotting the equation

Using a graphing utility, plot the polar equation \(r=\frac{4}{1+e \sin \theta}\) for values of e as \(e=0.1\), \(e=0.25\), \(e=0.5\), \(e=0.75\), and \(e=0.9\). Observe changes in the plotted graphs.
02

Identify and Discuss

Identify the form of the conic section represented by the plotted graphs. As \(e \rightarrow 1^{-}\), the conic transitions from circle (for \(e=0\)) to ellipse and eventually approaches to parabola (for \(e=1\)). As \(e \rightarrow 0^{+}\), the graph of conic section approaches to a circle.
03

Plotting for \(e=1\)

Plot the polar equation for \(e=1\). The graph is a parabola.
04

Plotting for values \(e>1\)

Plot the equation for \(e=1.1\), \(e=1.5\), and \(e=2\). These graphs represent a hyperbola.
05

Discuss changes for values \(e> 1\)

As \(e\rightarrow 1^{+}\), the graph transitions from a parabola to a hyperbola. As \(e \rightarrow \infty\), the hyperbola becomes a straight line.

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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 curves obtained from the intersection of a plane and a double-napped cone. The resulting shapes vary depending on the angle and position of the intersection. The four main types of conic sections are:
  • Circle - If the plane cuts perpendicular to the axis of the cone.
  • Ellipse - When the cut is at an angle, but not enough to reach the base of the cone.
  • Parabola - Formed when the plane is parallel to the tangent plane of the cone.
  • Hyperbola - When the plane intersects both nappes of the cone.
Conic sections are fundamental in mathematics as they describe a variety of real-world phenomena, from planetary orbits to optical properties. Understanding their properties and equations helps in visualizing and solving complex geometrical problems.
Graphing Utility
A graphing utility is a tool, often a calculator or a software application, that aids in visualizing mathematical equations. Here鈥檚 why it鈥檚 essential in dealing with polar equations:
  • Allows for a quick and accurate depiction of intricate curves.
  • Makes it easier to study the behavior of equations under different conditions, like changing eccentricity in conic sections.
  • Provides interactive capabilities to modify parameters and instantly observe results.
Using a graphing utility helps students understand the transition of shapes, such as how ellipses slowly turn into parabolas or hyperbolas based on the eccentricity. It is a practical method to bridge the gap between theoretical equations and their graphical representations.
Eccentricity
Eccentricity, denoted as "e", plays a crucial role in defining the shape of a conic section. Here鈥檚 a breakdown:
  • Circle: Eccentricity = 0. The graph is a perfect circle, symmetrical around a point.
  • Ellipse: 0 < eccentricity < 1. The conic forms a stretched circle, or ellipse, the closer to 1, the more elongated.
  • Parabola: Eccentricity = 1. The curve opens infinitely with no bounds.
  • Hyperbola: Eccentricity > 1. The graph splits into two curves, opening in opposite directions.
As eccentricity changes, the graph shows different behaviors. For instance, as "e" approaches 1 from below, the ellipse transitions into a parabola. As "e" moves past 1, the shape becomes a hyperbola, representing vastly different mathematical properties.
Polar Graph
A polar graph represents equations using a system where each point is defined by a radius and an angle, rather than Cartesian coordinates. For a given polar equation, such as \(r=\frac{4}{1+e \sin \theta}\), understanding the graph involves:
  • Interpreting the relationship between "r" (distance from the pole) and "\(\theta\)" (angle).
  • Noticing how alterations in eccentricity \(e\) affect the overall graph shape.
  • Being able to visualize changes without reliance on Cartesian planes.
Polar graphs provide a unique perspective for visualizing conic sections, as their structure naturally aligns with circular and radial patterns. This offers a different method for examining symmetry, orientation, and dimensions in mathematical problems.

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Most popular questions from this chapter

Sketch a graph of the polar equation. $$ r=4(1+\cos \theta) $$

Sketch a graph of the polar equation. $$ r=3-2 \cos \theta $$

Sketch a graph of the polar equation. $$ r \equiv 2 $$

Explain the process of sketching a plane curve given by parametric equations. What is meant by the orientation of the curve?

Arc Length find the arc length of the curve on the given interval. $$ \begin{array}{ll} \underline{\text { Parametric Equations }} & \underline{\text { Interval }} \\\ x=t^{2}, \quad y=2 t &\quad 0 \leq t \leq 2\end{array} $$
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