Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 6 - Section 6.4 - Graphs of Polar Equations - Exercise Set - Page 754: 54


To test the symmetry with respect to the line, replace $\left( r,\theta \right)$ with $\left( -r,-\theta \right)$, if the resultant equation is equivalent to the given equation, then the graph is symmetric with respect to $\theta =\frac{\pi }{2}$.

Work Step by Step

For example: We will replace $\left( r,\theta \right)\,\text{with}\,\left( -r,-\theta \right)$ in the polar equation $r=1-2\cos \theta $ to perform the test for the line $\theta =\frac{\pi }{2}$, $\begin{align} & r=1-2\cos \theta \\ & -r=1-2\cos \left( -\theta \right) \\ & -r=1-2\cos \theta \end{align}$ The resultant equation changes and thus it fails this symmetry test.
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