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 symmetric about the polar axis because replacing $\theta$ with $-\theta$ results in the same equation, indicating symmetry. It is symmetric about the pole because replacing $r$ with $-r$ also yields the same equation due to the squared term on the left side. Lastly, it is symmetric about the line $\theta = \pi / 2$ because replacing $\theta$ with $\pi - \theta$ results in the cosine of a double angle, which is symmetric about this line. Therefore, the equation exhibits all three types of symmetry.

Test the Polar Equation for Symmetry with Respect to the Polar