(572, #77) (a) In Example 11 the graphs suggest that the limacon has an inner loop when |c|>1. Prove that this is true, and find the values of that correspond to the inner loop.
(b) From Figure 18 it appears that the limacon loses its dimple when c = ½. Prove this.

Solution:

When |c| < 1, say c = ½, then , so the curve is a simple loop.

c = 0.5

When c = 1 and r = 0 for , which gives the dimple found in the graph.

c = 1


When |c| > 1, we take, for simplicity, c = 2. Then we have , and r = 0 when . In addition r < 0 for . These negative values of r give the inner loop.

c = 2

(a) From the graphs for different values of c, we see that for |c|>1 the limacon has an inner loop. To prove this analytically use the fact that r = 0 at the beginning and end of the loop. Then


This is not possible for |c|<1. Therefore |c|>1.

Maple code

by: nl