In the coordinate plane, circle C has radius r and its center is at po

Question

In the coordinate plane, circle C has radius r and its center is at point (x, y). Is at least 50% of the area of circle C contained in a single quadrant of the coordinate plane?

(1) x > y > 0

(2) x > r

Source: Advanced Quant Manhattan Prep 2020

TestPrepUnlimited · Answer

It would be hard to check how much % of the area is within a single quadrant, so we should jump to the statements to see if there's anything obvious.

Statement 1: 
x > y > 0, this tells us the circle is in Quadrant I but since we have no information on  r  this is insufficient.

Statement 2: 
The radius is less than x, then the entire circle is on the right side of the y-axis (note we can assume x is positive as r is positive). Thus, 100% of the circle is in either Quadrant I or Quadrant IV. Then no matter how you distribute this 100%, one of the Quadrants must have 50% of the circle or more. Sufficient.

Ans: B

Dip050701 · Answer

IanStewart  please help?

IanStewart · Answer

I don't think you'll see a question like this on the test, so I wouldn't really worry about it, but it's much easier to see what's happening here if you draw diagrams. If, using only Statement 1, you draw a circle with center (2, 1), and a radius of 1,000,000, you'll see that roughly 1/4 of the circle will be in each quadrant. But if the radius is 0.00001, then the entire circle will be in the first quadrant, so Statement 1 is not sufficient.

Statement 2 is the more difficult statement. You might first think of an example using concrete numbers, where x > r -- say you have a circle with its center at (7, 2) and a radius of 4. Notice that the leftmost point on the circumference of the circle will be 4 units (the radius) to the left of the center, and the rightmost point will be 4 units to the right of the circle, so (3, 2) and (11, 2) will be the endpoints of a diameter of the circle. So in this case, none of the points on or inside the circle have a negative x-coordinate; that is, none of the points in the circle are in quadrants II or III. More than half of this circle is in quadrant I, and the rest is in quadrant IV. Now if we instead use the letters in the question, we have a circle centered at (x, y) with radius r. We know that (x - r, y) and (x + r, y) will be endpoints of one diameter of the circle. But Statement 2 tells us that x > r, or x - r > 0. So both of these points have a positive x-coordinate, and again, none of the points on or inside this circle can be in quadrants II or III. The circle is divided between quadrants I and IV only, so at least half of the circle must be in one of those two quadrants, and the answer is B.

bumpbot · Answer

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In the coordinate plane, circle C has radius r and its center is at po

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In the coordinate plane, circle C has radius r and its center is at po

Prep Toolkit

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