A few ways really. If you're just doing it by eye, this might work:
1: Draw the circle.
2: Create a vertical guideline intersecting the circle where you want it to end (it will intersect the circle at two points an equal distance from top and bottom).
3: Use the scissors tool to cut the circle at the two intersection points, and delete the surplus arc.
If it has to be 270 degrees exactly, you could alternatively:
1: Create the circle
2: Draw a square and rotate it 45 degrees.
3: Align the leftmost point of the square with the center of the circle (the Transform palette will allow you determine and align these points precisely)
4: Use the PathFinder palette to remove the square from the circle, and remove any surplus paths.
There's probably lots of other ways of doing this -- I don't have Illustrator in front of me right now, but these might get you started anyway!