Phase portraits - An (erroneous?) observation

[Matthew Pick]

Today while writing the quiz, I was attempting to figure out question 2 with little avail.

Grasping at straws - I decided to enter in 4 points into the formula for the derivative for x' (the points were (1,0), (-1,0), (0,1), (-1,0) )

I noticed that the slopes for (1,0) and (-1,0) were negative inverses.
The same was true for the slopes of (0,1) and (0,-1).
I guessed that this implied that each of the pairs of points were on circles (ellipses? in any case it was a centre..). Upon checking the answers on the forum this evening, this seems to be true.

Is this always the case? I realize that this is not a particularly rigorous way of solving, but it seems like it might be a good method to check answers if anyone, like I am, is prone to computational mistakes...

[Victor Ivrii]

Why you just don't learn a very simple criteria for the center or focus: if the top-right element of $2\times 2$ matrix is $>0$ the orientation is clock-wise; otherwi\se it is counter-clock-wise