(1) x^2+y^2=1. Recall that (x−y)2≥0 (square of any number is more than or equal to zero). Expand: x^2−2xy+y^2≥0 and since x2+y2=1 then: 1−2xy≥0. So, xy≤1/2. Sufficient.
(2) x^2−y^2=0. Re-arrange and take the square root from both sides: |x|=|y|. Clearly insufficient.
I am fine with (2). I have trouble with (1).
What if, instead of using x^2-2xy+y^2≥0, I decided to use x^2+
2xy+y^2≥0 (note the positive). That would result in xy≥-1/2 instead of the xy≤1/2 that is sufficient. In the end, I would have to use x^2-2xy+y^2≥0?
Anyone care to elaborate on this please?
Yes, to solve this question you should use (x−y)^2≥0 not (x+y)^2≥0.