Answer
a. symmetric with respect to the polar axis.
b. symmetric with respect to the line $\theta=\frac{\pi}{2}$.
c. symmetric with respect to the pole.
Work Step by Step
a. We are given the polar equation $r^2=4cos2\theta$. To test the symmetry with respect to the polar axis, let $\theta\to -\theta$; we have $r^2=4cos2(-\theta)=4cos2\theta$. Thus, the equation is symmetric with respect to the polar axis.
b. To test the symmetry with respect to the line $\theta=\frac{\pi}{2}$, let $r\to -r$ and $\theta\to -\theta$; we have $(-r)^2=4cos2(-\theta)$ or $r^2=4cos2\theta$. Thus, the equation is symmetric with respect to the line $\theta=\frac{\pi}{2}$.
c. To test the symmetry with respect to the pole, let $r\to -r$; we have $(-r)^2=4cos2\theta$ or $r^2=4cos2\theta$. Thus, the equation is symmetric with respect to the pole.
