Any question that asks about QUADRANTS of a COORDINATE PLANE is really asking about POSITIVE v. NEGATIVE.

Translate the given information:

If ab ≠ 0 --> neither of them is 0

points (-a,b) and (-b,a) are in the same quadrant of the xy-plane --> for the x coordinates -a and -b to be in the same quadrant, then a and b must have the same sign.
If they're both positive, then (-a, b) will be in Quadrant II (neg, pos).
If they're both negative, then (-a, b) will be in Quadrant IV (pos, neg).

Rephrase the question:
is point (-x,y) in this same quadrant? --> (-x, y) will be in the same quadrant as (-a, b) and (-b, a) if x and y both have the same sign as a and b.

(1) xy > 0

Translation --> x and y have the same sign as each other. (If their product is positive, they must be either both positive or both negative.)
This doesn't tell us if x and y have the same sign as a and b, though. Insufficient.

(2) ax > 0

Translation --> a and x have the same sign.
This doesn't tell us about y, so it's not enough to answer the question. Insufficient.

(1) & (2) together:

If x has the same sign as y, a has the same sign as x, and - as we know from the question stem - b has the same sign as a, then all 4 variables must have the same sign. (-x, y) will be in the same quadrant as (-a, b) and (-b, a). Sufficient.

The answer is C.

You're very welcome!

That intuition for what a question is REALLY asking is definitely something that you can build over time. Here are the 2 biggest things you can do to build it:

1. Don't dive in and start solving right away. Give yourself 20-30 seconds to think it through before your pen hits the paper. More on that here: https://www.manhattanprep.com/gmat/blog ... -pen-down/

2. Review each question in depth and record your takeaways. Don't just quickly scan the explanation and move onto the next problem. More on that here: https://www.manhattanprep.com/gmat/blog ... -studying/

Good luck!