Question

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

.$r=3\sec \theta$

Short answer

The polar equation$r=3\sec \theta$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 $r=3\sec \theta$.

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$r=3\sec \theta$.

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

Replacing$(r,\theta )=(-r,-\theta )$and simplifying the equation.
If the resulting equation $r=3\sec \theta$is equal to the original equation then it is symmetry about the line $\theta =\frac{\pi }{2}$.

$\begin{array}{l}-r=3\sec (-\theta )\\ -r=3\sec \theta \\ \text{r}=-3\sec \theta \end{array}$

The polar equation $r=3\sec \theta$ 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$r=3\sec \theta$then it is symmetry about the polar axis.

$\begin{array}{l}\text{r}=3\sec (-\theta )\\ \text{r}=3\sec \theta \end{array}$

The polar equation $r=3\sec \theta$ 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$r=3\sec \theta$then it is symmetry about the pole.

$\begin{array}{l}-r=3\sec \theta \\ \text{r}=-\left(3\sec \theta \right)\\ \text{r}=-3\sec \theta \end{array}$

The polar equation $r=3\sec \theta$ is not symmetric to the pole.