Question:medium

Is \(xy\) negative?

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

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Use the identity \( (x+y)^2-(x-y)^2=4xy \) to convert statement 1 directly into a sign for xy.
Updated On: Jul 21, 2026
  • If the data in statement (1) alone is sufficient to answer the question
  • If the data in statement (2) alone is sufficient to answer the question
  • If the data in both the statements together are needed to answer the question
  • If either statement (1) alone or statement (2) alone is sufficient to answer the question
Show Solution

The Correct Option is A

Solution and Explanation

Step 1: Expand statement 1 directly, without shortcuts.
$(x+y)^2 = x^2 + 2xy + y^2$
$(x-y)^2 = x^2 - 2xy + y^2$
Statement 1 says $(x+y)^2 < (x-y)^2$, so $x^2 + 2xy + y^2 < x^2 - 2xy + y^2$.

Step 2: Simplify to isolate xy.
Cancel $x^2$ and $y^2$ from both sides to get $2xy < -2xy$.
Add $2xy$ to both sides to get $4xy < 0$.
Divide by 4 to get $xy < 0$, so xy is negative for certain. Statement 1 alone answers the question.

Step 3: Check statement 2 with sample numbers.
Statement 2 only fixes that $x - y$ is positive.
Try $x = 5$, $y = 2$: here $x - y = 3 > 0$ and $xy = 10$, positive.
Try $x = 2$, $y = -5$: here $x - y = 7 > 0$ but $xy = -10$, negative.
Both examples obey statement 2 yet give opposite signs for xy, so statement 2 alone cannot decide the question.

Final Answer:
Only statement 1 alone settles whether xy is negative. \[ \boxed{\text{Option (a): Statement 1 alone is sufficient}} \]
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