Friday, September 10, 2010

Is x^2 + y^2 > 4a?

(1) (x + y)^2 = 9a
(2) (x – y)^2 = a




Official Answer to the above problem.

(1) INSUFFICIENT: If we multiply this equation out, we get:

x2 + 2xy + y2 = 9a

If we try to solve this expression for

x2 + y2, we getx2 + y2 = 9a – 2xy

Since the value of this expression depends on the value of x and y, we don't have enough information.

(2) INSUFFICIENT: If we multiply this equation out, we get:

x2 – 2xy + y2 = a

If we try to solve this expression for x2 + y2,
we getx2 + y2 = a + 2xy

Since the value of this expression depends on the value of x and y, we don't have enough information.

(1) AND (2) INSUFFICIENT: We can combine the two expanded forms of the equations from the two statements by adding them:

x2 + 2xy + y2 = 9ax2 – 2xy + y2 = a----- 2x2 + 2y2 = 10ax2 + y2 = 5a

If we substitute this back into the original question, the question becomes: "Is 5a > 4a?"If a > 0, the answer is yes.We know from the question stem that a is nonnegative.However, if a = 0 the answer is no.

The correct answer is E.

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