Question

Is \(xy\) negative?

Statement 1: \((x+y)^2 < (x-y)^2\)
Statement 2: \((x-y)\) is positive

• A. If the data in statement (1) alone is sufficient to answer the question
• B. If the data in statement (2) alone is sufficient to answer the question
• C. If the data in both the statements together are needed to answer the question
• D. If either statement (1) alone or statement (2) alone is sufficient to answer the question

Hint:

Use the identity \( (x+y)^2-(x-y)^2=4xy \) to convert statement 1 directly into a sign for xy.

Answer 1

The Correct Option is A

Step 1: Expand statement 1.
The identity \( (x+y)^2 - (x-y)^2 = 4xy \) holds for all real x and y.
Statement 1 says \( (x+y)^2 < (x-y)^2 \).
This means \( (x+y)^2 - (x-y)^2 < 0 \).

Step 2: Translate the inequality into a sign for xy.
Using the identity, \( (x+y)^2 - (x-y)^2 = 4xy \).
So the inequality becomes \( 4xy < 0 \).
Dividing both sides by 4 gives \( xy < 0 \).
This directly answers the question: xy is negative. Statement 1 alone is sufficient.

Step 3: Test statement 2 alone.
Statement 2 only says \( (x-y) \) is positive, so x is greater than y.
This gives no information about the signs of x and y individually.
For example, x = 3, y = 1 gives xy = 3, positive. x = 1, y = -3 gives xy = -3, negative.
Both cases satisfy statement 2, so the sign of xy cannot be pinned down. Statement 2 alone is not sufficient.

Final Answer:
Statement 1 alone answers the question, statement 2 alone does not. \[ \boxed{\text{Option (a): Statement 1 alone is sufficient}} \]