
Concept explainers
To check: Check whether the given statement is true or false.

Answer to Problem 8AYU
Solution:
It is a false statement.
Given:
Statement says that the tests for symmetry in polar coordinates are always conclusive
Explanation of Solution
Test for symmetry:
Symmetry with respect to the polar axis (-Axis):
In a polar equation, replace by . If an equivalent equation results, the graph is symmetric with respect to the polar axis.
Symmetry with respect to the line :
In a polar equation, replace by . If an equivalent equation results, the graph is symmetric with respect to the line .
Symmetry with respect to the Pole:
In a polar equation, replace by or by . If an equivalent equation results, the graph is symmetric with respect to the line .
The three tests for symmetry given here are sufficient conditions for symmetry, but they are not necessary conditions. That is, an equation may fail these tests and still have a graph that is symmetric with respect to the polar axis, the line , or the pole. For example, the graph of turns out to be symmetric with respect to the polar axis, the line , and the pole, but only the test for symmetry with respect to the pole (replace by works.
Therefore, the tests for symmetry in polar coordinates need not to be conclusive always.
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