/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 76 Graph each polar equation. $$ ... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 76

Graph each polar equation. $$ r=\frac{1}{1-\cos \theta} \quad(\text {parabola}) $$

Short Answer

Expert verified
The graph is a parabola extending indefinitely outward for \( \theta = 0\) and forming a curve for other values of \(\theta\).

Step by step solution

01

Identify the type of conic

The given equation is in the form of a polar equation for a conic section, specifically a parabola. The standard form of a parabola in polar coordinates is given by \ r = \frac{k}{1 - e\cos\theta} with \(e = 1\).
02

Recognize parameter values

here, compare the given equation \( r = \frac{1}{1 - \cos\theta} \) with the standard form. We can see that \(e=1\), indicating it's a parabola, and k is also 1.
03

Plot Points

To graph a polar equation, determine coordinates by substituting values of \(\theta\) into the equation. Use values covering important points between 0 and 2\(\pi\) e.g., 0, \(\pi/2\), \(\pi\), 3\(\pi/2\), and 2\(\pi\).
04

Evaluate and plot specific points

Evaluate points:\(\theta=0\), \(r = \frac{1}{1-\cos 0}=\frac{1}{0} \rightarrow \infty\)\ \(\theta = \pi/2\), \(r = \frac{1}{1-0}=1 \)\(\theta=\pi\), \(r = \frac{1}{1-(-1)}=\frac{1}{2}\)\(\theta=3\pi/2\), \(r = \frac{1}{1-0}=1\)
05

Draw the Graph

Using the points calculated in Step 4, plot them in polar coordinates and sketch the parabola. Remember that for \( \theta = 0\) , \( r \) tends to infinity, indicating the graph extends indefinitely in that direction.

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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 by intersecting a plane with a cone. Based on the angle and position of the intersection, four types of conic sections can be produced: circles, ellipses, parabolas, and hyperbolas. Each of these has unique properties and equations.
The Parabolas Concept
A parabola is a specific type of conic section defined as the set of all points that are equidistant from a fixed point called the focus and a line called the directrix. In polar coordinates, a parabola is characterized by its eccentricity (\(e\)) equal to 1. The general equation of a parabola in polar form is given by: \[ r = \frac{k}{1 - \, e \cos \theta} \]. Here, k is a constant, and for parabolas, e = 1. Therefore, the equation simplifies to \[ r = \frac{k}{1 - \cos\theta} \]. By comparing the given equation \( r = \frac{1}{1 - \, \cos \theta} \), we see that it represents a parabola with k = 1.
Graphing Polar Coordinates
Graphing polar coordinates involves plotting points based on their distance from the origin (radius, \(r\)) and the angle from the positive x-axis (\(\theta\)). To graph the polar equation \( r = \frac{1}{1 - \, \cos \theta} \), follow these steps:
  • Substitute different values of \(\theta\) (e.g., 0, \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), 2\(\pi\)) to find corresponding \(r\) values.
  • For \(\theta = 0\), \( r = \frac{1}{1 - \, \cos 0} \) results in \(r\) being infinite, indicating the graph extends indefinitely in that direction.
  • For \(\theta = \frac{\pi}{2}\), \( r = \frac{1}{1 - \, \cos \frac{\pi}{2}} \) equals 1, providing the point (1, \(\frac{\pi}{2}\)).
  • For \(\theta = \pi\), \( r = \frac{1}{1 - (-1)} = \frac{1}{2} \), giving the point (0.5, \(\pi\)).
  • For \(\theta = \frac{3\pi}{2}\), \( r = \frac{1}{1 - 0} \) equals 1 again.
After calculating these points, plot them on polar coordinates and connect the dots to sketch the parabola.

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Most popular questions from this chapter

A 2-pound weight is attached to a 3 -pound weight by a rope that passes over an ideal pulley. The smaller weight hangs vertically, while the larger weight sits on a frictionless inclined ramp with angle \(\theta .\) The rope exerts a tension force \(\mathbf{T}\) on both weights along the direction of the rope. Find the angle measure for \(\theta\) that is needed to keep the larger weight from sliding down the ramp. Round your answer to the nearest tenth of a degree.

If \(z=2-5 i\) and \(w=4+i,\) find \(z \cdot w\).

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=4 \mathbf{i}-2 \mathbf{j}\)

Decompose \(\mathbf{v}\) into two vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\), and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\). $$ \mathbf{v}=3 \mathbf{i}+\mathbf{j}, \quad \mathbf{w}=-2 \mathbf{i}-\mathbf{j} $$

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ 4 x^{2} y=1 $$
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