Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r ^ { 2 } = 9 \cos 2 \theta$
I $\text { symmetric about the polar axis }$
II $\text { symmetric about the pole }$
III $\text { symmetric about the line } \theta = \pi / 2$
A)I only
B)I and II
C) I and III
D)II and III
E) I, II, III

Question

Quizplus · Accepted Answer

To test for symmetry in polar coordinates:- Symmetry about the polar axis ($\theta \rightarrow -\theta$): Replace $\theta$ with $-\theta$. Since $\cos(-2\theta) = \cos(2\theta)$, the equation remains unchanged, indicating symmetry about the polar axis.- Symmetry about the pole ($r \rightarrow -r$): Since the equation involves $r^2$, replacing $r$ with $-r$ does not change the equation, indicating symmetry about the pole.- Symmetry about the line $\theta = \pi/2$ ($\theta \rightarrow \pi - \theta$): Replace $\theta$ with $\pi - \theta$. Since $\cos(2(\pi - \theta)) = \cos(2\pi - 2\theta) = \cos(-2\theta) = \cos(2\theta)$, the equation remains unchanged, indicating symmetry about the line $\theta = \pi/2$.Thus, the equation is symmetric about the polar axis, the pole, and the line $\theta = \pi/2$.

Test the Polar Equation for Symmetry with Respect to the Polar