Chapter 10.3, Problem 68E

(a)

To determine

To plot: The graph of $r=\frac{2}{1+e\cos \theta }$ for $e=0.1$ , $0.3$ , $0.5$ , $0.7$ and $0.9$ .

Answer to Problem 68E

The graph of $r=\frac{2}{1+e\cos \theta }$ for $e=0.1$ , $0.3$ , $0.5$ , $0.7$ and $0.9$ 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 $e=0.1$ , $0.3$ , $0.5$ , $0.7$ and $0.9$ .

Calculation:

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

Draw the graph of $r\left(e\right)$ for $e=0.1$ , $0.3$ , $0.5$ , $0.7$ and $0.9$ on the same plane as shown below.

  

Figure (1)

From the graph it can be observed that the graph of $r=\frac{2}{1+e\cos \theta }$ for $e=0.1$ , $0.3$ , $0.5$ , $0.7$ and $0.9$ are ellipses.

Therefore, the graphs of $r=\frac{2}{1+e\cos \theta }$ for $e=0.1$ , $0.3$ , $0.5$ , $0.7$ and $0.9$ is an ellipse shown in Figure (1).

(b)

To determine

To explain: A conjecture for the graphs of the curve $r=\frac{2}{1+e\cos \theta }$ for $0<e<1$ and $e\to 0^{+}$ .

Answer to Problem 68E

The graph of the curve $r=\frac{2}{1+e\cos \theta }$ for $0<e<1$ is an ellipse and the graph approaches the circle with radius $2$ and center at the origin for $e\to 0^{+}$ .

Explanation of Solution

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

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

From part (a), the graph of $r=\frac{2}{1+e\cos \theta }$ for $e=0.1$ , $0.3$ , $0.5$ , $0.7$ and $0.9$ is an ellipse.

From above observation it can be concluded that the graph of the curve $r=\frac{2}{1+e\cos \theta }$ for $0<e<1$ is an ellipse.

The graph of $r=\frac{2}{1+e\cos \theta }$ approaches the circle with radius $2$ and center at the origin as $e\to 0^{+}$ .