Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r ^ { 2 } = \cos 2 \theta$
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

The given polar equation is $r^2 = \cos 2\theta$. To test for symmetry:- Symmetry about the polar axis ($\theta$ to $-\theta$): Substituting $-\theta$ for $\theta$, we get $r^2 = \cos 2(-\theta) = \cos(-2\theta) = \cos 2\theta$, which is the same as the original equation. So, it is symmetric about the polar axis.- Symmetry about the pole (origin) ($r$ to $-r$): Since the equation is in terms of $r^2$, replacing $r$ with $-r$ does not change the equation ($(-r)^2 = r^2$). Thus, it is symmetric about the pole.- Symmetry about the line $\theta = \pi/2$ ($\theta$ to $\pi - \theta$): Substituting $\pi - \theta$ for $\theta$, we get $r^2 = \cos 2(\pi - \theta) = \cos(2\pi - 2\theta) = \cos(-2\theta) = \cos 2\theta$, which is the same as the original equation. Therefore, it is also symmetric about the line $\theta = \pi/2$.Hence, the equation is symmetric with respect to the polar axis, the pole, and the line $\theta = \pi / 2$.

Test the Polar Equation for Symmetry with Respect to the Polar