Explanation

Put X = x/a, Y = y/b: then X + Y = a + b and X/a + Y/b = 2. Substituting Y = a+b−X into the second gives X/a = 1, so X = a and Y = b. Then x/a² − y/b² = X/a − Y/b = 1 − 1 = 0. Hence option C.