Question

Testing for Symmetry Test the polar equation for symmetry to the polar axis, the pole, and the line$\theta =\pi /2$

$r^2=4\cos 2\theta$.

Short answer

The polar equation $r^2=4\cos 2\theta$ is symmetric to the polar axis, the pole, and the line$\theta =\frac{\pi }{2}$ .

Step-by-step solution

Step 1. Given information.

The given polar equation $r^2=4\cos 2\theta$.

Step 2. Finding given equation Symmetric or not.

If the polar equation$r^2=4\cos 2\theta ,(r,\theta )$can be replaced

by$(r,-\theta )$or$(-r,\pi -\theta )$, then the graph is symmetric to the pole.

If the polar equation$r^2=4\cos 2\theta ,(r,\theta )$can be replaced by$(-r,\theta )$or$(r,\pi +\theta )$, then the graph is symmetric to the pole.

If the polar equation$r^2=4\cos 2\theta ,(r,\theta )$can be replaced by$(r,\pi -\theta )$or$(-r,-\theta )$ , then the graph is symmetric to the line$\theta =\frac{\pi }{2}$.

The given polar equation is$r^2=4\cos 2\theta$

Testing for the symmetry of the line$\theta =\frac{\pi }{2}$.

Replacing$(r,\theta )=(-r,-\theta )$and simplifying the equation

If the resulting equation$r^2=4\cos 2\theta$is equal to the original equation then it is symmetry about the line$\theta =\frac{\pi }{2}$ .

$\begin{array}{l}\text{Invalid <m:msup> element}=4\cos (-2\theta )\\ r^2=4\cos 2\theta \end{array}$

The polar equation $r^2=4\cos 2\theta$ is not symmetric to the line $\theta =\frac{\pi }{2}$.

Step 3. Testing for symmetry about page polar axis.

Replacing$(r,\theta )=(r,-\theta )$ and simplifying the equation $r^2=4\cos 2\theta$ then it is symmetry about the polar axis.

$\begin{array}{c}{r}^2=4\cos (-2\theta )\\ r^2=4\cos 2\theta \end{array}$

The polar equation $r^2=4\cos 2\theta$ is symmetric to the polar axis.

Step 4. Testing for symmetry about the pole.

Replacing$(r,\theta )=(-r,\theta )$and simplifying the equation

If the resulting equation is equal to the original equation$r^2=4\cos 2\theta$ is symmetric to the pole.

$\begin{array}{c}{(-r)}^2=4\cos 2\theta \\ r^2=4\cos 2\theta \end{array}$

The polar equation $r^2=4\cos 2\theta$ is symmetric to the pole.