Intersection of ellipses and equivalent problems

[Gerenuk]

Does anyone know how to determine whether two ellipses intersect? I don't need the precise points but rather only the answer whether there are points. All my attempts led to 4th order polynomials, which are heavy to solve, but considering that I don't need the actual points I assume there must be an easier way.

Some guy claims it's doable
http://www.cut-the-knot.org/htdocs/dcforum/DCForumID6/710.shtml

Here are some equivalent problems which have to be solved for the angles (which however I can't solve either...)
\cos\phi+a\sin\theta=x
\sin\phi+b\cos\theta=y
or even
\Re(e^{i\theta}(1+ze^{i\theta}))=q
where z is complex, is an equivalent problem. Any ideas?

My best attempt so far was using discriminants, but it's messy and I made a mistake somewhere...

 

[fresh_42]

I would eliminate the angles by squaring and write the equations in Cartesian coordinates. Then we have still a quadratic equation in two variables, which one can be normalized by a coordinate transformation, such that we end up with $y=\pm \frac{b}{a}\sqrt{a^2-x^2}$ with one of the two ellipsis. This can be substituted into the other equation, such that we have only one equation in $x$. It is probably easiest to determine it numerically by one the tools available on the internet.