How to create a line curving around a circle

Create the first circle; use Object > Path > Offset Path to make another circle outside of it. 

Move the inner circle to a new layer above, lock it and turn off its visibilty. Draw a Stroke across the outer circle, and use Divide Objects Below to cut the circle. Delete the bottom section, then use the scissors tool to cut the bottom line of the remaining shape. Grab the center point, and move each side in turn to the outside, constraining the angle wtih the shift key. 

 Turn the inner circle back on in the layers panel. Select the outer shape, then select the a corner control and drag it until you are happy with the curve.