# Intersection of ellipses and equivalent problems

1. Jul 10, 2010

### 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...