An idealized model of the path of a moon (relative to the Sun) moving with constant speed in a circular orbit around a planet, where the planet in turn revolves around the Sun, is given by the parametric equations

x(θ) = a cos θ + cos nθ, y(θ) = a sin θ + sin nθ. 

The distance from the moon to the planet is taken to be 1, the distance from the planet to the Sun is a, and n is the number of times the moon orbits the planet for every 1 revolution of the planet around the Sun. Plot the graph of the path of a moon for the given constants; then conjecture which values of n produce loops for a fixed value of a.

a. a = 4, n = 3.

b. a = 4, n = 4.

c. a = 4, n = 5.