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Chapter 13: Problem 20

Sketch the graph of the polar equation. $$ r(3 \cos \theta-2 \sin \theta)=6 $$

Short Answer

Expert verified
The graph of the given polar equation is a line with slope \(\frac{3}{2}\) and y-intercept \(-3\).

Step by step solution

01

Convert the Equation to Rectangular Form

Start with the polar equation \( r(3 \cos \theta - 2 \sin \theta) = 6 \). Using the conversions: \( x = r \cos \theta \) and \( y = r \sin \theta \), replace \( r \cos \theta \) with \( x \) and \( r \sin \theta \) with \( y \). Thus, the equation becomes:\[ 3x - 2y = 6 \].
02

Rearrange for Slope-Intercept Form

Rearrange the equation \( 3x - 2y = 6 \) to express \( y \) in terms of \( x \). Solve for \( y \):\[ 2y = 3x - 6 \]\[ y = \frac{3}{2}x - 3 \].
03

Identify the Slope and Y-Intercept

From the equation \( y = \frac{3}{2}x - 3 \), identify the slope and y-intercept. The slope is \( \frac{3}{2} \) and the y-intercept is \( (0, -3) \).
04

Sketch the Line

Plot the y-intercept \( (0, -3) \) on a rectangular coordinate system. Use the slope \( \frac{3}{2} \), which means for every 3 units you move up, you move 2 units to the right. Draw the line through these points.

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Key Concepts

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

Rectangular Coordinates
Rectangular coordinates, often known as Cartesian coordinates, are a foundational concept in geometry. They allow us to specify points on a plane using a pair of numerical values:
  • The x-coordinate (or abscissa), which measures horizontal distance from the origin.
  • The y-coordinate (or ordinate), measuring vertical distance from the origin.
Each point in this coordinate system is denoted as \( (x, y) \). The system makes it easy to visualize mathematical equations, such as lines, parabolas, and circles, by plotting points on the xy-plane.
When we convert polar equations to rectangular form, as in the exercise, we use the relationships: \ x = r \cos \theta \, and \ y = r \sin \theta \.
This transformation allows us to easily graph more complex equations by making them linear in respect to x and y.
Slope-Intercept Form
The slope-intercept form of a linear equation is an efficient way to express a line in terms of its slope and y-intercept.
  • The slope indicates the steepness and direction of the line, calculated as the ratio of the vertical change to the horizontal change between two points on the line.
  • The y-intercept is the point where the line crosses the y-axis, typically expressed as \( (0, b) \).
A linear equation in slope-intercept form appears as \ y = mx + b \, where \ m \ is the slope and \ b \ is the y-intercept. In our exercise, after converting the polar equation to rectangular form, the equation \ 3x - 2y = 6 \ was rearranged into this form as \ y = \frac{3}{2}x - 3 \. This makes it easier to understand and plot on a graph since you immediately see the slope \ \frac{3}{2} \ and y-intercept \ (-3) \.
Graphing Polar Equations
Graphing polar equations involves plotting points and curves in the polar coordinate system, which is different from the rectangular or Cartesian coordinate system. In polar coordinates, each point on a plane is determined by:
  • The distance from the origin known as the radius \( r \).
  • The angle \ \theta \, measured from a fixed direction, typically the positive x-axis.
In this task, the equation \ r(3 \cos \theta - 2 \sin \theta) = 6 \ begins in polar form. However, by transforming it into the rectangular form, we simplify graphing as it's then treated like a standard linear equation in Cartesian coordinates.
The benefit of such transformations is the ability to leverage familiar techniques for line graphing in algebra, while maintaining the flexibility of analyzing circular patterns unique to polar systems.

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Most popular questions from this chapter

Sketch the graph of the polar equation. $$ r=4-4 \sin \theta $$

If the graphs of the polar equations \(r=f(\theta)\) and \(r=g(\theta)\) intersect at \(P(r, \theta),\) prove that the tangent lines at \(P\) are perpendicular if and only if $$ f^{\prime}(\theta) g^{\prime}(\theta)+f(\theta) g(\theta)=0 $$ (The graphs are said to be orthogonal at \(P\).)

Find the slope of the tangent line to the graph of the polar equation at the point corresponding to the given value of \(\theta\). $$ r=8 \cos 3 \theta ; \quad \theta=\pi / 4 $$

Sketch the graph of the polar equation. $$ r=3 \sin 2 \theta $$

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. $$ r=2 \cos \theta+3 \sin \theta $$
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