I found this page which gave me some equations on solving the intersection of a line with an ellipse given a point on the line and the slope of the line:
There Isn't much explanation but I presume that after solving for $a$, $b$, and $c$, you can then find the roots to the newly formed quadratic, which will give two possible $k$ values.
Here's where the questions come in:
- Which of the two $k$ values do I use in solving for $r$ and $z$?
- How can you tell when the line doesn't intersect the ellipse? Is the quadratic equation for $k$ not have any real roots?
Earlier, the site states:
and that 
Is it important for $r$ to be greater than zero in for this equation even though I'm not testing a point, but a line instead?
Why is the semi-minor axis being defined as $a_e(1-f)$? I usually just define the semi-minor axis the same way I define the major, so could I just replace all the $(1-f)$'s in the above equations with my desired semi-minor axis length?
Finally, is there any simpler, faster way to see if and were an ellipse and line intersect?