Chapter 11.3, Problem 68E

(a)

To determine

To graph: The curve $r$ for $k=0.1,0.3,0.5,0.7,0.9$ .

Answer to Problem 68E

The curve is an ellipse.

Explanation of Solution

Given information:

The curve: $r=\frac{2}{1+k\cos \theta }$

Calculation:

The given curve is $r=\frac{2}{1+k\cos \theta }$ .

For values of $k=0.1,0.3,0.5,0.7,0.9$ ; the graph of the given curve is shown below:

  

From the above graph it is seen that the curve is an ellipse.

(b)

To determine

To determine: The conjecture about the graphs for $0<k<1$ and $k\to 0^{+}$ .

Answer to Problem 68E

The graph moves towards the circle whose radius is 2 and the center is the origin for $k\to 0^{+}$ .

Explanation of Solution

Given information:

The curve: $r=\frac{2}{1+k\cos \theta }$

Calculation:

The given curve is $r=\frac{2}{1+k\cos \theta }$ .

For values of $k=0.1,0.3,0.5,0.7,0.9$ ; the graph of the given curve is shown below:

  

From the above graph it is seen that the curve is an ellipse.

As $k\to 0^{+}$ , the graph moves towards the circle whose radius is 2 and the center is the origin.