Question

In Problems 8 and 9, test the polar equation for symmetry with respect to the pole, the polar axis, and the line $\theta =\frac{\pi }{2}$.

$r^2\cos \theta =5$

Short answer

The polar equation is symmetric about the polar axis, the pole and the line $\theta =\frac{\pi }{2}$.

Step-by-step solution

Step 1. Given information  

We are given a polar equation,

$r^2\cos \theta =5$

We have to check the polar equation for symmetry with respect to the pole, the polar axis, and the line $\theta =\frac{\pi }{2}$.

Step 2. Checking the symmetry about polar axis

We have to replace $\theta$with $-\theta$to check the symmetry,

localid="1647058749450" $$\begin{gathered} r^2\cos \left(\theta \right)=5 \\ {r}^2\cos (-\theta )=5 \end{gathered}$$

We know that cosine function is even function, so

$r^2\cos \left(\theta \right)=5$

Hence, the function is symmetric about polar axis.

Step 3. Checking the symmetry about pole

We have to replace r with $-r$to check the symmetry,

$$\begin{gathered} r^2\cos \left(\theta \right)=5 \\ {\left(-r\right)}^2\cos \left(\theta \right)=5 \\ {r}^2\cos \left(\theta \right)=5 \end{gathered}$$

Hence, the function is symmetric about the pole.

Step 4. Checking the symmetry about the line θ=π2

Since the function is symmetric about both polar axis and origin, it has to be symmetric about the line $\theta =\frac{\pi }{2}$

Hence, the given polar equation is symmetric about the polar axis, the pole and the line $\theta =\frac{\pi }{2}$.