The question is asked in Visual Complex Analysis by Tristam Needham Chapter 2 Exercise 7. It refers to the orthogonal trajectories we drew in the previous exercise to a set of Cassinian curves. I worked out that they were hyperbolae and found a (rather non-standard) equation. But I do not understand what 'significance' can signify in this context.
I suspect that this lack of understanding points to a significant gap in my understanding of the whole topic. Hints would be most welcome.
I found the answer browsing through pictures of Cassinian curves. The orthogonal trajectories to the family show the magnetic flux as it would appear were an equal current to be passed through the two foci.