If xy≠0, and x-y= x- y, which of the following must be true? i) xy=1 ii) x>0 iii) x^2-2xy+y^2=0

If xy≠0, and x-y= x- y, which of the followin… | GMAT Sprint

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705-805

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If $xy≠0$ , and $\sqrt{x-y}=\sqrt{x}-\sqrt{y}$ , which of the following must be true? i) $\frac{x}{y}=1$ ii) $x>0$ iii) $x^2-2xy+y^2=0$

Answer Choices

Correct answer marked below

i only

i & ii only

i & iii only

ii & iii only

i, ii & iii

Correct

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Explanation

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Official Explanation

Official

Given:

\sqrt{x-y}=\sqrt{x}-\sqrt{y}

Square both sides:

x - y = x + y - 2\sqrt{xy}

Simplify:

2\sqrt{xy} = 2y

Divide by 2:

\sqrt{xy} = y

Square both sides again:

xy = y^2

Rearrange:

y(x - y) = 0

Since

xy≠0

, we know

y≠0

, so

x - y = 0

.

Thus,

x = y

.

From

x = y

:…

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What this question tests

What it tests

Your handling of square and higher roots, including simplifying radicals.

Common trap

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.

If xy≠0xy≠0xy=0, and x−y=x−y\sqrt{x-y}=\sqrt{x}-\sqrt{y}x−y​=x​−y​, which of the following must be true? i) xy=1\frac{x}{y}=1yx​=1 ii) x>0x>0x>0 iii) x2−2xy+y2=0x^2-2xy+y^2=0x2−2xy+y2=0