Hi, there! I'm happy to help with this.
First, a quick review of quadrants: what defines the quadrants are the +/- signs of x and y
1) In Quadrant I, x > 0 and y > 0
2) In Quadrant II, x < 0 and y > 0
3) In Quadrant III, x < 0 and y < 0
4) In Quadrant VI, x > 0 and y < 0
If (-a, b) and (-b, a) are in the same quadrant, that means that the x-coordinates have the same sign, and also the y-coordinates have the same sign. Look at the y-coordinates --- if the two points are in the same quadrant, a & b have the same sign. They either could both be positive (in which case, the points would be in Quadrant II) or they could both be negative (in which case, the points would be in Quadrant IV).
Now, the question is: (-x, y) in the same quadrant as these two points?
(1)
Statement 1: xy > 0
This tells us that x and y have the same sign --- both positive or both negative. Now, we know a & b have the same sign, and x & y have the same sign, but there's two possibilities for each, so we don't know whether a & b & x & y all have the same sign. This is insufficient.
(2)
Statement 2: ax > 0
This, by itself, tells us that a and x have the same sign -- with this alone, we know that a & b & x all have the same sign, but we have zeor information about y. This too is insufficient.
Combined (1) & (2)Prompt tells us a & b have the same sign. Statement #1 tells us x & y have the same sign. Statement #2 tells us x & a have the same sign. Put it all together --> we now know that x & y & a & b all have the same sign. Therefore, (-x, y) will have the same sign x- & y-coordinates as (-a, b) & (-b, a), and therefore all will be in the same quadrant. Combined statements are sufficient.
Answer = C
Here's another coordinate plane practice question just for practice.
http://gmat.magoosh.com/questions/1028Does all that make sense? Please let me know if you have any additional questions.
Mike
_________________
Mike McGarry
Magoosh Test PrepEducation is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)