Question

In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{-1}{1-\cos \theta}\)

Short answer

The graph of the given equation is a circle with a diameter along the negative x-axis.

Step-by-step solution

Understanding the Polar Equation

In the polar equation \(r=\frac{-1}{1-\cos \theta}\), \(r\) is the distance of the point from the origin, while \(\theta\) is the angle made by the line segment joining the origin and the point with the positive x-axis. As \(\theta\) changes, the value of \(r\) will change accordingly. To visualize this graph, it's necessary to relate the values of \(r\) and \(\theta\).

Use a Graphic Tool

Using a graphing utility such as Desmos or GeoGebra, plot the given polar equation. Since \(\theta\) ranges from 0 to 2\(\pi\), make sure the graphing utility is set to a similar interval, this will ensure the entire graph is generated.

Identify The Graph

Once the curve is plotted, identify the shape. For this equation, the graph produced is a circle with a diameter along the negative x-axis. This occurs because the values of \(r\) and \(\theta\) are inversely related, causing the graph to wrap around the origin with a period of 2\(\pi\).