Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:

(A) Quantity A is greater.

(B) Quantity B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.

A circle with radius 5 and passing through the origin cuts off intercepts m and n with the x and y axes respectively. Let “k” be the number of values m*n can take.

Quantity A: k

Quantity B: 3

**Answer: (B)**

**Explanation:**

This is based on a simple idea that a circle passing through the origin that cuts off intercepts with the coordinate axes will form a right-angled triangle with origin as its vertex forming the right angle. Hence the hypotenuse of the right triangle so formed will be the diameter of its circumcircle.

So if radius is 5 then the diameter is 10. Consequently with 10 as the hypotenuse of a right angled triangle the other two sides can only be 6 and 8 as these are supposed to be integers. Therefore, the value of k can be 6*8 = 48.

However, since the intercepts can be cut off on the negative axis also we can have -48 also as a value for k.

P.S: Thinking of it, there seems to be one more possibility of a circle that passes through origin and with radius 5 as shown below.

However, this circle doesn’t “cut off” any intercept with the x axis although we do get x = 0 as the value for the x-intercept when we substitute y = 0 in the equation of the circle x

^{2}+ y^{2}– 10y = 0 or x^{2 }+ y^{2 }+ 10y = 0. If that be the case, then ‘0’ is also a possible value for k and the answer to this question would be (C). While it is understandable that there is an ambiguity with this problem you can be assured that you are less likely to face similar issues in your actual GRE.
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