Chapter 10.3, Problem 70E

(a)

To determine

To plot: The graph of $r=\frac{2}{1+e\cos \theta }$ for $e=1.1$ , $1.3$ , $1.5$ , $1.7$ and $1.9$ .

Answer to Problem 70E

The graph of $r=\frac{2}{1+e\cos \theta }$ for $e=1.1$ , $1.3$ , $1.5$ , $1.7$ and $1.9$ is an ellipse shown in Figure (1).

Explanation of Solution

Given information: The given equation is $r=\frac{2}{1+e\cos \theta }$ . The given values are $e=1.1$ , $1.3$ , $1.5$ , $1.7$ and $1.9$ .

Calculation:

Consider the given equation $r=\frac{2}{1+e\cos \theta }$ .

Draw the graph of $r\left(e\right)$ for $e=1.1$ , $1.3$ , $1.5$ , $1.7$ and $1.9$ on the same plane as shown below.

  

Figure (1)

From the graph it can be observed that the graphs of $r=\frac{2}{1+e\cos \theta }$ for $e=1.1$ , $1.3$ , $1.5$ , $1.7$ and $1.9$ is hyperbola.

Therefore, the graph of $r=\frac{2}{1+e\cos \theta }$ for $e=1.1$ , $1.3$ , $1.5$ , $1.7$ and $1.9$ is a hyperbola shown in Figure (1).

(b)

To determine

To explain: A conjecture for the graph of the curve $r=\frac{2}{1+e\cos \theta }$ for $e>1$ and $e\to 1^{+}$ .

Answer to Problem 70E

The curve $r=\frac{2}{1+e\cos \theta }$ is a hyperbola for $e>1$ and the left branches of the curve $r=\frac{2}{1+e\cos \theta }$ approach a parabola when $e\to 1^{+}$ .

Explanation of Solution

Given information: The given equation is $r=\frac{2}{1+e\cos \theta }$ .

From part (a), the graph of $r=\frac{2}{1+e\cos \theta }$ for $e>1$ is a hyperbola.

From the graph in part (a), it can be observed that as $e\to 1^{+}$ the left branches of the hyperbolas approach a parabola. As $e\to 1^{+}$ the right branches of the hyperbolas approach to infinity.

Therefore, the curve $r=\frac{2}{1+e\cos \theta }$ is a hyperbola for $e>1$ and the left branches of the curve $r=\frac{2}{1+e\cos \theta }$ approach a parabola when $e\to 1^{+}$ .