Formula for line-ellipse collisions?

Question

So I'm making a game and it requires odd line-ellipse collisions, as well as line-line collisions. I've found a solution for line-line collisions, but haven't found a formula for line-ellipse collisions. Help?
Here is a image from my game, the line there collides with the stick body but not the head: https://www.sololearn.com/post/479441/?ref=app
More info on the game: https://www.sololearn.com/post/479439/?ref=app
Edit: circle-line collisions will do, it's an even circle, no rotation
Edit 2: just a formula will do, thanks for your time!

Answer 1

Let's make it simpler.
What do you want to achieve?

Answer 2

RKK I'm trying to find a formula for line-ellipse collisions