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What is the value of x^2+y^2?

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What is the value of x^2+y^2?

by Vincen » Tue Jan 09, 2018 6:15 am
$$\text{what}\ \text{is}\ \text{the}\ \text{value}\ \text{of}\ x^2+y^2?$$ $$\left(1\right)\ \ x^2+y^2=2xy+1\ .$$ $$\left(2\right)\ \ x^2+y^2=4-2xy.$$ The OA is the option C.

Experts, can you show me how to use each statement to get an answer here? Please.
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Source: — Data Sufficiency |
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by DavidG@VeritasPrep » Tue Jan 09, 2018 9:23 am
Vincen wrote:$$\text{what}\ \text{is}\ \text{the}\ \text{value}\ \text{of}\ x^2+y^2?$$ $$\left(1\right)\ \ x^2+y^2=2xy+1\ .$$ $$\left(2\right)\ \ x^2+y^2=4-2xy.$$ The OA is the option C.

Experts, can you show me how to use each statement to get an answer here? Please.
Statement 1: $$x^2 -2xy + y^2 = 1$$
$$(x-y)^2 = 1$$
Clearly not sufficient.
Case 1: x = 3 and y = 2 --> x^2 + y^2 = 9 + 4 = 13
Case 2: x=5 and y = 4. --> x^2 + y^2 = 25 + 16 = 41

Statement 2: $$x^2 + 2xy + y^2 = 4$$
$$(x + y)^2 = 4
Clearly not sufficient
Case 1: x = 0 and y = 2 --> x^2 + y^2 = 0 + 4 = 4
Case 2: x=1 and y = 1. --> 1^2 + 1^2 = 1 + 1 = 2

Together, we can sum the equations
x^2 + y^2 = 2xy + 1 and
x^2 + y^2 = 4-2xy
_____________________
2x^2 + 2y^2 = 5
x^2 + y^2 = 5/2. A unique value. Sufficient. The answer is C
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