Question

Use a graphing utility to graph the hyperbolas \(r=\frac{e}{1+e \cos \theta},\) for \(e=1.1,1.3,1.5,1.7\) and 2 on the same set of axes. Explain how the shapes of the curves vary as \(e\) changes.

Short answer

Answer: As the value of e increases, the distance from the center of the hyperbola to the pole (origin) increases and the shapes of the hyperbolas become more flattened and elongated along the horizontal axis.

Step-by-step solution

Graphing the polar equation

First, use a graphing utility of your choice (such as Desmos, Wolfram Alpha, or a graphing calculator) to plot the given polar equation \(r=\frac{e}{1+e \cos \theta}\) for the following values of \(e\): 1.1, 1.3, 1.5, 1.7, and 2. Plug each value of \(e\) into the polar equation and plot each curve on the same set of axes. For example, when \(e=1.1\), the polar equation is \(r=\frac{1.1}{1+1.1 \cos \theta}\). Plot this curve and then repeat the process for the remaining values of \(e\).

Observing and analyzing the changes in the curves

Now, observe the shapes of the resulting curves for each value of \(e\). Notice the following characteristics: - As \(e\) increases, the distance from the center of the hyperbola to the pole (origin) also increases. - With the increase of \(e\), the shapes of the hyperbolas become more "flattened" and their branches are more elongated along the horizontal axis. To conclude, as \(e\) changes, the shapes and positions of the hyperbolas vary. Specifically, the distance from the center of the hyperbola to the pole increases, and the shape becomes more flattened and elongated along the horizontal axis.