Problem 64 - Calculus Early Transcendentals

Question

Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as \(\theta\) increases from 0 to \(2 \pi\). $$r=\frac{1}{1-2 \cos \theta}$$

Step-by-Step Solution

Answer

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.

Step 1: Understanding Polar Coordinates and Conversions

Polar coordinates are a coordinate system where each point in the plane is identified by its distance, r, from a reference point, known as the pole, and the angle \(\theta\) formed between the line connecting the pole and the point, and a reference axis, known as the polar axis. To convert from polar coordinates to Cartesian coordinates, the following equations are used: $$x = r\cos\theta$$ $$y = r\sin\theta$$

Step 2: Determine the Graph in Cartesian Coordinates

Start by writing the given polar equation in Cartesian coordinates using the conversion equations: $$r = \frac{1}{1 - 2\cos\theta}$$ $$x = r\cos\theta$$ $$y = r\sin\theta$$ Replacing r in x and y equations using the given polar equation: $$x = \frac{\cos\theta}{1 - 2\cos\theta}$$ $$y = \frac{\sin\theta}{1 - 2\cos\theta}$$ Multiplying both numerator and denominator of the equations by the conjugate of the denominator, \((1 + 2\cos\theta)\), we get: $$x = \frac{\cos\theta (1 + 2\cos\theta)}{1 - 4\cos^2\theta}$$ $$y = \frac{\sin\theta (1 + 2\cos\theta)}{1 - 4\cos^2\theta}$$ Recall that \(\sin^2\theta + \cos^2\theta = 1\). Therefore, we can replace \(\cos^2\theta\) with \((1 - \sin^2\theta)\): $$x = \frac{\cos\theta (1 + 2\cos\theta)}{1 - 4(1 - \sin^2\theta)}$$ $$y = \frac{\sin\theta (1 + 2\cos\theta)}{1 - 4(1 - \sin^2\theta)}$$ The following equations for x and y are obtained after simplification, and we can recognize the graph as a hyperbola: $$x = \frac{\cos\theta (1 + 2\cos\theta)}{-3 + 4\sin^2\theta}$$ $$y = \frac{\sin\theta (1 + 2\cos\theta)}{-3 + 4\sin^2\theta}$$

Step 3: Show the Points and Curve Generation

To show how the curve is generated as \(\theta\) increases from 0 to \(2\pi\), we will choose some key points, which are equally spaced, and show their polar coordinates along with their corresponding Cartesian coordinates, using the equations obtained above. The key points chosen are: \(\theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}\). Next, use these values to calculate their corresponding r and Cartesian coordinates using the equations provided. Then draw the curve, marking these points and the arrows to indicate how the curve is generated. Make sure to connect the arrows and points in the order in which they appear as \(\theta\) increases from 0 to \(2\pi\). By doing that and drawing the curve, it will become clear how the curve is generated by following the arrows and labeled points as \(\theta\) increases in the given range.