a) pi is no good. That's six intersections. You must keep it under 2.51 or so. It would be a nice challenge to get exactly 4 intersections.
b) Try just a little over 5.6. Really, you just have to figure out how to solve using these equations. I'm kind of wondering why the tangent is in a "conic section" problem. Again, finding exactly 8 intersections would be a good problem.
c) It would help me if you specified the exact "nature" of the tangent. Specifically, tan(x) = -tan(-x). This property has a name. What is it? In other words, if you hit y = tan(x) at (a,b), you MUST hit it again at (-a,-b), a reflection across the Origin.