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Chapter 11: Problem 20

Graph the sets of points whose polar coordinates satisfy the equations and inequalities. $$\theta=\pi / 2, \quad r \leq 0$$

Short Answer

Expert verified
The graph is a single point at the origin (0,0).

Step by step solution

01

Understand Polar Coordinates

Polar coordinates consist of a distance, \( r \), from a central point (the origin), and an angle \( \theta \), which is the direction expressed in radians measured counterclockwise from the positive x-axis.
02

Interpret the Given Condition for \( \theta \)

The equation \( \theta = \frac{\pi}{2} \) specifies that the angle is directly pointing upwards along the positive y-axis. In polar coordinates, this means any point satisfying this condition lies along the vertical line through the y-axis.
03

Interpret the Given Condition for \( r \)

The inequality \( r \leq 0 \) means that the distance from the origin to the point is zero or negative. Since distance cannot be negative, \( r \leq 0 \) effectively means that \( r = 0 \) is the only possible distance satisfying this, as it implies the point is located at the origin.
04

Determine the Graphical Representation

Since \( r = 0 \) and \( \theta = \frac{\pi}{2} \), the set of points satisfying these conditions consists only of the origin because a negative radius doesn't make sense for any actual position. Thus, the graphical representation is just the origin, a single point at (0,0) in a Cartesian plane.

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Key Concepts

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

Angle in Radians
In polar coordinates, angles play a pivotal role in defining the direction of a point relative to the origin. Here, we measure angles using a unit called radians, which is an alternative to degrees.
While degrees part the circle into 360 sections, radians are a mathematical constant that divides the circle into approximately 6.28 parts, or 2\( \pi \) radians.
For instance:
  • 0 or 2\( \pi \) radians point right along the positive x-axis.
  • \( \frac{\pi}{2} \) radians point upwards along the positive y-axis, just as in our problem.
  • 3\( \frac{\pi}{2} \) radians would point downward along the negative y-axis.
Understanding radians allows us to succinctly express rotational positions and directions on a plane. These angular measures are essential for graphing and locating points in polar coordinates, as they guide us in determining where a point will lie relative to the origin.
Distance from Origin
The concept of distance in polar coordinates is uniquely defined by the radius, denoted as \( r \).
This variable \( r \) indicates how far away a point is from the origin. When plotting, we consider \( r \) in conjunction with the angle \( \theta \).
It's important to note:
  • The distance, or radius, \( r \) is typically a non-negative real number. Positive values imply a position along the defined angle.
  • A value of zero means the point is at the origin itself.
  • Negative values theoretically imply a distance the opposite direction of \( \theta \), though this isn't applied in standard polar graphing.
In our exercise, \( r \leq 0 \) leads us to conclude that only \( r = 0 \) is permissible. This condition solidifies that the point remains at the origin, as no other points satisfy the condition of a valid distance when \( r \) can't be negative.
Graphical Representation
Visualizing polar coordinates on a graph involves understanding both angle and distance to plot the exact position of a point.
Unlike Cartesian coordinates that use x and y, polar coordinates use \( r \) and \( \theta \) to give meaning to the location of points.
Let's break down the graphical representation:
  • To graph an angle, start from the positive x-axis and move counterclockwise to \( \theta \). For \( \theta = \frac{\pi}{2} \), point upwards directly along the y-axis.
  • The distance \( r \) tells "how far" to move along the heading produced by \( \theta \). For our exercise, \( r = 0 \) implies no movement from the origin.
  • Since no movement is possible beyond the origin, our graphical set of points reduces to a mere point at (0,0) in the Cartesian plane.
This result showcases the simplicity of polar coordinates when restrictions limit the distance to zero. The graphical representation clearly indicates the singular point location, reinforcing our interpretation.

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Most popular questions from this chapter

Find a polar equation in the form \(r \cos \left(\theta-\theta_{0}\right)=r_{0}\) for each of the lines. $$y=-5$$

Graph the lines and conic sections. $$r=1 /(1+2 \cos \theta)$$

Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. $$r=2 \cos \theta+2 \sin \theta$$

Sketch the lines and find Cartesian equations for them. $$r \cos \left(\theta+\frac{3 \pi}{4}\right)=1$$

Graph the lines and conic sections. $$r=-2 \cos \theta$$
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