Chapter 6.5, Problem 44E

Solution

To analyze: The graph given by the polar equation $r^2=9\cos 2\theta$ such that $0\le \theta \le 2\pi$ .

The equation represents a lemniscate. Domain of the graph is $\left[0,\frac{\pi }{4}\right]\cup \left[\frac{3\pi }{4},\frac{5\pi }{4}\right]\cup \left[\frac{7\pi }{4},2\pi \right]$ . The range is $\left[0,3\right]$ . The graph is continuous and symmetric about both the axes and the origin. Maximum $r$ value is $3$ . There are no asymptotes to the graph.

Given information:

The given polar equation is $r^2=9\cos 2\theta$ .

Calculation:

Rewrite the polar equation as, $r=\pm 3\sqrt{\cos 2\theta }$ .

Plot the graph for the polar equation.

  

Figure (1)

From the figure (1), it can be observed that the equation represents a lemniscate.

The equation is defined only when $\cos 2\theta \ge 0$ . Therefore, the domain of the graph is $\left[0,\frac{\pi }{4}\right]\cup \left[\frac{3\pi }{4},\frac{5\pi }{4}\right]\cup \left[\frac{7\pi }{4},2\pi \right]$ .

The range of $\cos 2\theta$ , is $0\le \cos 2\theta \le 1$ . Therefore, the range of $r^2=9\cos 2\theta$ is $\left[0,3\right]$ by taking maximum value of $\cos 2\theta$ as $1$ and minimum value as $0$ .

The graph is continuous on its domain. The graph is symmetric about both the axes and the origin.

Since $r$ values are bounded by $\left[0,3\right]$ , so the graph is bounded.

The maximum value of $r$ is $3$ .

There are no asymptotes to the graph.

Therefore, the polar equation $r^2=9\cos 2\theta$ is analyzed with the help of figure (1).