Problem 25

Given information

 Investigate the family of curves given by the parametric equations. In a particular determine the value of for which there is a loop and find the point where the curve intersect itself. What happens to the loop as increases? Find the coordinates of the leftmost and the rightmost point of the loop.

Step-by-step explanation

Given:

The graph for the families of parametric equation for all the values of c 

Here 

From the graph it is seen that when the is positive the graph intersect itself at a point on axis having the coordinates

The leftmost and the rightmost both have vertical tangents having slope

Then 

Now for

Thus the rightmost point is

Similarly 

Now for

Thus the leftmost point is