91Ó°ÊÓ

Question: Graph the polar equation $$r = \frac{1}{1 - 2\cos\theta}$$ and show how the curve is generated as \(\theta\) increases from 0 to \(2\pi\). Answer: To graph the polar equation and show how the curve is generated, first convert the polar equation into Cartesian coordinates using the conversion equations. The resulting equations for x and y are: $$x = \frac{\cos\theta (1 + 2\cos\theta)}{-3 + 4\sin^2\theta}$$ $$y = \frac{\sin\theta (1 + 2\cos\theta)}{-3 + 4\sin^2\theta}$$ These equations represent a hyperbola. Next, choose key points equally spaced in the range of \(\theta\) from 0 to \(2\pi\), such as \(\theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}\) and calculate their corresponding r and Cartesian coordinates. Finally, draw the curve with these points, labeling them according to their \(\theta\) values and indicating the direction of curve generation with arrows. As \(\theta\) increases from 0 to \(2\pi\), the curve is generated by following the arrows and labeled points in the order they appear.