Suppose that the position of one particle at time $t$ is given by $$x_1 = 3\sin(t), \quad y_1 = 2\cos(t), \quad 0 \le t \le 2\pi$$ and the position of a second particle is given by $$x_2 = -3+\cos(t), \quad y_2 = 1+\sin(t), \quad 0 \le t \le 2\pi.$$

(a) How many points of intersection are there for these paths?

Number of intersection points =

(b) How many of these points of intersection are collision points? In other words, how many times are the particles in the same place at the same time?

Number of collision points =

(c) List the collision points. List them in terms of increasing $x$-coordinates. If several collision points have the same $x$-coordinate, list those in order of increasing $y$-coordinates. Type an upper-case "N" in each unused answer blank.

First collision point: $(x, y)$ = (, )

Second collision point: $(x, y)$ = (, )

Third collision point: $(x, y)$ = (, )

(d) Now suppose that the path of the second particle is given by $$x_2 = 3+\cos(t), \quad y_2 = 1+\sin(t), \quad 0 \le t \le 2\pi$$ and the path of the first particle remains the same. How many points of intersection are there for these paths?

Number of intersection points =

(e) For the situation described in (d), how many of these points of intersection are collision points?

Number of collision points =