help visualizing the shape

[EnriqueR905]

Hello,i got this question in my last CAT.However,i cannot understand how to draw the shape and understand what XY=0 is supposed to be used for.

A circle is drawn in the xy-coordinate plane. If there are n different points (x, y) on the circle such that xy = 0, then the possible values of n are


0, 1, or 2

0, 2, or 4

0 or 4

0, 2, 3, or 4

0, 1, 2, 3, or 4

the answer is (E) and this is the explanation:

For xy to equal zero, one or both of the coordinates of n need to equal zero. This is only true for points on the axes. So the real question here is “How many points could this circle have on the x- and y-axes?”

0: If we draw a circle entirely in a single quadrant, there will be no points touching the axes, so xy will never equal zero.

1: If we draw a circle that is only tangent to one axis (i.e. touching it, rather than crossing over it), xy will only equal zero at that one point.

2: There are multiple ways to have two points where xy = 0. The circle could be tangent to both axes, or it could cross through one axis in two places.

3: Again, there are multiple ways for this to occur, but the easiest to visualize is a circle that is tangent to one axis, but which passes through the other axis twice.

4: Any circle that is built around the origin in such a way that it passes through all four axes will have four points where xy = 0.

THANK YOU.

[RonPurewal]

think about when a product is equal to 0. this happens whenever ANY number in the product is 0.

so, "xy = 0" means that EITHER x OR y (or both) is 0.

so, "a point with xy = 0" is a point anywhere on either of the coordinate axes.

in other words, the question is asking how many different times a circle can touch the coordinate axes.

that can happen 0, 1, 2, 3, or 4 times. make sure you can draw each of these cases.