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Chapter 8: Problem 597

For the following exercises, test each equation for symmetry. $$r=7$$

Short Answer

Expert verified
The equation \( r = 7 \) is symmetric about the polar axis only.

Step by step solution

01

Identify the Equation Type

The given equation is in polar form: \( r = 7 \). This means it represents a circle centered at the origin with a radius of 7.
02

Test for Symmetry with Respect to the Polar Axis

To test for symmetry about the polar axis (the line \( \theta = 0 \)), replace \( (r, \theta) \) with \( (r, -\theta) \). The equation remains \( r = 7 \) indicating symmetry about the polar axis.
03

Test for Symmetry with Respect to the Line \( \theta = \frac{\pi}{2} \)

Replace \( (r, \theta) \) with \( (-r, \pi - \theta) \). The equation simplifies to \( -r = 7 \), which contradicts the original. Thus, it is not symmetric with respect to the line \( \theta = \frac{\pi}{2} \).
04

Test for Symmetry with Respect to the Origin

Replace \( (r, \theta) \) with \( (-r, \theta + \pi) \). This leads to \( -r = 7 \), which does not satisfy the original equation. Therefore, it is not symmetric with respect to the origin.

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Key Concepts

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

Symmetry in Polar Equations
In polar equations, symmetry plays a crucial role in understanding the shape and properties of a graph. Symmetry in this context refers to the invariance of a polar equation when certain transformations are applied. These transformations subtly shift or reflect points within the coordinate system yet leave some equations unchanged.

Testing for symmetry can be quite methodical and involves manipulating the polar coordinates \(r\) and \(\theta\):
  • Polar Axis Symmetry: Replace \((r, \theta)\) with \((r, -\theta)\).
  • Line \(theta = \frac{\pi}{2}\) Symmetry: Replace \((r, \theta)\) with \((-r, \pi - \theta)\).
  • Origin Symmetry: Replace \((r, \theta)\) with \((-r, \theta + \pi)\).
Successfully identifying symmetry simplifies sketching and comprehending these equations.
Circle Equations
Polar equations that simplify to expressions like \(r = a\) represent circles. These are among the simplest graphical forms, with \(a\) defining the circle's radius. The equation \(r = 7\), for example, describes a circle centered at the origin with a radius of 7.

In polar coordinates, the interpretation of \(r = a\) is straightforward:
  • The distance from the origin to any point on the circle is constant (\

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