First you should consider applications of dragging and 'straight lines' to decie which of the two paths will fit the rear wheel and which will fit the front wheel. (If in doubt, just watch how the front and rear wheels of a bus move when it turns. Or you can get out your bike, get the wheels muddy and ride through Vari Hall!)
Consider how the bicycle wheels sit along the tracks at any time: they are tangent to the two curves. Now consider the frame of the bike - it is a direct extension of the tangent vector for the back wheel - so it is a straight line segment extending out from the back wheel to the point just above where the front wheel is tangent to the other curve. Draw this.
With this in mind, given any position of the rear wheel, and a selected direction of travel, you know where the front wheel is at the same time. (Extend the tangent vector till it crosses the other curve.) If you move this rear wheel / direction picture along one curve you see a series of measurements for the length of the crossbar on the bike. Is it constant? If it is not, you are looking at an impossible path for the bike.
Can you now find a direction (and a choice of back wheel / front wheel path ) which works?
Notice that I do not need to examine the depth of the tracks etc. It is pure geometry with the diagram you are given.