A comet has an elliptical orbit that is $144$ billion miles across the $x$-axis and $48$ billion miles across on the $72$ years to complete one revolution. If the center of the coordinate system is the center of the ellipse, find a set of parametric equations that describes this motion. Include an interval for $t$ that starts at $t=0$ and finishes when one revolution has been completed.
A planet in the same coordinate system is centered at $(72,4)$ is $24$ billion miles across on the axis parallel to the $x$ axis and $8$ billion miles across on the axis parellel to the $y$ axis. It completes one revolution every $4$ years. Find a set of parametric equations that describes this motion. Include an interval for $t$ that starts at $t=0$ and finishes when one revolution has been completed.
The two orbits intersect. How many points of intersection are there? Find the points of intersection.
Solution attempt:
No idea, any help is appreciated
You can model this using an ellipse. Since it needs to be parametric, use the $\cos$ and $\sin$ functions to model it. Create models for both ellipses, and then set them equal to each other. Then solve.