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Chapter 10: Problem 46

Sketch the graph of the polar equation. $$r=-2$$

Short Answer

Expert verified
The graph of \( r = -2 \) is a circle of radius 2, centered at the pole, with points plotted in the opposite direction.

Step by step solution

01

Understanding the Polar Equation

A polar equation describes the relationship between the radius \( r \) and the angle \( \theta \) in polar coordinates. In this case, the equation \( r = -2 \) specifies that the radius is \(-2\), which means the point is located 2 units in the opposite direction from the pole along the given angle.
02

Identifying the Graph Type

Since \( r = -2 \) is a constant value, we can identify this polar graph as a circle with a fixed radius of 2. However, because the radius is negative, each point on this circle would be plotted 2 units in the opposite direction from where a positive \( r = 2 \) point would be.
03

Plotting Points in Polar Coordinates

In polar coordinates, every point can be represented as \((r, \theta)\). For \( r = -2 \), this implies that for every angle \( \theta \) (ranging from \( 0 \) to \( 2\pi \)), we plot the point in the direction opposite to the angle \( \theta \) at a distance of 2 units from the pole.
04

Drawing the Graph

The graph will be a circle centered at the pole. The circle is drawn with radius 2, but since \( r = -2 \), it will appear as a circle shifted through the pole. All points are effectively diametrically opposite their equivalent positions if \( r \) were positive.

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Key Concepts

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

Polar Equation
A polar equation provides a way to describe the relationship between the radius and the angle using polar coordinates. Polar coordinates are particularly useful for scenarios involving circular or rotational symmetry. An example of a polar equation can be as simple as \( r = c \), where \( r \) is the radius and \( c \) is a constant value, either positive or negative.
This form simplifies the visualization of shapes like circles, spirals, or complex curves. With \( r = -2 \) as in the example, the radius is a constant, but negative, suggesting a reflection through the pole for all plotted angles \( \theta \).
Radius
The radius in polar coordinates determines how far a point is from the pole, which is the center or origin of the polar coordinate system. Typically, it's denoted by \( r \).
When \( r \) is positive, the point is located at distance \( r \) away from the pole, in the direction dictated by \( \theta \).
  • Positive \( r \): Points away from the pole.
  • Negative \( r \): Points in the opposite direction of \( \theta \), at distance \( |r| \).
In our example, \( r = -2 \) means that rather than moving outward, we move opposite to the typical direction and place the point 2 units across the pole.
Angle
In polar systems, the angle \( \theta \)鈥攐ften measured in radians鈥攍ocates the direction from the pole. It's similar to how the hands of a clock relate the position over time, providing orientation for where \( r \) will extend or contract.
  • \( \theta = 0 \): Points directly to the right on standard polar graphs.
  • \( \theta = \pi/2 \): Points upwards.
  • \( \theta = \pi \): To the left.
  • \( \theta = 3\pi/2 \): Points downwards.
Within our given equation, \( r = -2 \), for every angle \( \theta \), the point's position is noted by extending in the direction opposite to \( \theta \), at a fixed distance of 2.
Graph of Polar Equations
Graphing polar equations involves plotting all possible points defined by the equation onto polar coordinate axes. Each point corresponds to a pair \((r, \theta)\). Given our polar equation, \( r = -2 \), the graph forms a unique circle.
This circle, unlike one with a positive radius, appears shifted because every point is plotted in the direction opposite its angle. This concept means that all the resulting points are diametrically opposed to what would be a traditional circle of radius 2. Look on your graph for a complete circle centered at the pole, reflecting how every negative radius forms a mirrored outward movement.

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Most popular questions from this chapter

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of Intersection. $$\left\\{\begin{array}{l} x^{2}+4 y^{2}=36 \\ x^{2}+y^{2}=12 \end{array}\right.$$

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