I wonder if there might be some typographical error in the statement of your problem. Simply assign a fixed value to a and to b. Then P and Q will be fixed points which determine a fixed line. If you replace c by the variable x then R is a an oval-shaped curve, so you can be certain that for most choices of a, b, and c, the points will not be collinear. (Also, since b never appears alone, you can simplify the way the problem looks by replacing a-b everywhere with d.)

Chris