Question

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

$r=4+8\cos \theta$.

Short answer

The polar equation $r=4+8\cos \theta$ is not symmetric to the pole and the line $\theta =\frac{\pi }{2}$.

Step-by-step solution

Step 1. Given information.

The given polar equation, $r=4+8\cos \theta$.

Step 2. Finding given equation Symmetric or not.

If the polar equation$r=4+8\cos \theta ,(r,\theta )$can be replaced by$(r,-\theta )$or$(-r,\pi -\theta )$, then the graph is symmetric to the pole.

If the polar equation$r=4+8\cos \theta ,(r,\theta )$can be replaced by$(-r,\theta )$or$(r,\pi +\theta )$, then the graph is symmetric to the pole.

If the polar equation$r=4+8\cos \theta ,(r,\theta )$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$r=4+8\cos \theta$

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

Replacing$(r,\theta )=(-r,-\theta )$and simplifying the equation.
If the resulting equation $r=4+8\cos \theta$is equal to the original equation then it is symmetry about the line. $\theta =\frac{\pi }{2}$

$\begin{array}{l}-r=4+8\cos (-\theta )\\ -r=4+8\cos \theta \\ \text{r}=-(4+8\cos \theta )\\ \text{r}=-4-8\cos \theta \end{array}$

The polar equation $r=4+8\cos \theta$ 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

If the resulting equation is equal to the original equation$r=4+8\cos \theta$then it is symmetry about the polar axis.

$\begin{array}{c}r=4+8\cos (-\theta )\\ r=4+8\cos \theta \end{array}$

The polar equation $r=4+8\cos \theta$ 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$r=4+8\cos \theta$then it is symmetry about the pole.

$\begin{array}{c}-r=4+8\cos \theta \\ \text{r}=-(4+\cos \theta )\\ \text{r}=-4-\cos \theta \end{array}$

The polar equation $r=4+8\cos \theta$is not symmetric to the pole.