Re: the two flight paths, think of the satellite as the static leg of a drawing compass. The 'pencil' of the compass draws a circle around that 'leg'. They know something 'pinged' off the circle, but do not know where on the circle it pinged.
What they do know is that some parts of the circle can be ruled out, either because too far away to have been reached in that time, or because of overlap with radar/too close to last known position. That leaves two parts of a circle - the arcs - uneliminated.