If a is nonnegative, is x2 + y2 > 4a?

[suneelv001]

If a is nonnegative, is x2 + y2 > 4a?

(1) (x + y)2 = 9a

(2) (x - y)2 = a

QA:C
But my question is, it should be E, as 'a' can be zero or +ve. Please advise.

[santanu.bn]

I) X^2 + Y^2 + 2xy = 9a
thus X^2 + Y^2 = 9a - 2xy

We don't know what is xy in terms of a. INSUFFICIENT - Cancel A and D

II)X^2 + Y^2 - 2xy = a
thus X^2 + Y^2 = a + 2xy

We don't know what is xy in terms of a. INSUFFICIENT - Cancel B

Combining I and II

9a - 2xy = a + 2xy
So 4xy = 8a
So 2xy = 4a

Thus using I or II solution is possible. Sufficient - Answer C
a is Zero or +ve. But i believe it shouldn't matter in this case.

[mschwrtz]

You're right, if x, y, and a are all 0, then both statements are true, a is nonnegative, and the answer to the question is "NO." For any other values of x y and a legal for both statements, the answer is "YES."

If the two statements together allow both yes and n, then they're not sufficient.

BUT...are you sure that the original problem did not contain any constraints on x and y?

[marc.gagnon]

The original problem (challenge problem 06/05/2006) does not contain any additional constraints, other than the stated constraint of: "If a is nonnegative"

BUT...are you sure that the original problem did not contain any constraints on x and y?

I have a follow up question regarding this problem:
I have retraced the solution step numerous times, but I must be missing something simple. I understand the part I highlighted in Red to say that since x^2 + y^2 is the sum of squares, and squares are never negative, then the sum must be positive, as a result the 9a must be positive, as 9a = x^2 + y^2. Please stop me here, if I am offtrack.

I get lost at the next statement, which I highlighted in blue.
Is the 'inequality', referenced in the solution below, the question stem "(x^2 + y^2 > 4a)" and the previous equation, referenced below, statement (1). I have highlighted statement (1) in Red?
- If these are not the areas being referenced by the solution statement, then which inequality/statement are being referred to in the solution?
- If these are the items being referenced
(a) how can we include the question stem in the proof?
(b) How do we arrive at 2(x^2 + y^2) >= 9a

MGMAT Solution
(1) INSUFFICIENT: If we multiply this equation out, we get:
x^2 + 2xy + y^2 = 9a
If we try to solve this expression for x^2 + y^2, we get
x^2 + y^2 = 9a - 2xy
Since the value of this expression depends on the value of x and y, we don't seem to have enough information.

As a subtlety, we do know that
x^2 + y^2 >= 2xy
in all cases, because x^2 + y^2 - 2xy = (x - y)^2, which must be non-negative. Thus, if we add this inequality to the previous equation, we get the following:
2(x^2 + y^2) >= 9awhich means
x^2 + y^2 >= 4.5a

And thus, the question can be rephrased as follows:

Is an expression at least as big as 4.5a greater than 4a?

The answer is Yes, if we know a is positive. However, a could be 0, in which case the answer would be Maybe.

[jnelson0612]

Marc, I have read this several times and am still not sure what you are asking. Are you still here and wondering about this?