DS question

[hemant_rajput]

77. What is the value of x^2 + y^2 ?
(1) x^2 + y^2 = 2xy + 1
(2) x^2 + y^2 = 4 - 2xy
[spoiler]
OA:E[/spoiler]
Source: Jeff Seckman

[srcc25anu]

IMO it should be C

ST1: x^2 + y^2 = 2xy + 1
we cannot find 3 variables from 1 eqaution
Hence Insufficient

St2: x^2 + y^2 = 4 - 2xy
we cannot find 3 variables from 1 eqaution
Hence Insufficient

Together: Adding the 2 equations we get 2(x^2 + y^2) = 1 + 4 (xy terms cancel out)
Hence x^2 + y^2 = 5/2
Hence Sufficient

[hemant_rajput]

IMO it should be C

ST1: x^2 + y^2 = 2xy + 1
we cannot find 3 variables from 1 equation
Hence Insufficient

St2: x^2 + y^2 = 4 - 2xy
we cannot find 3 variables from 1 equation
Hence Insufficient

Together: Adding the 2 equations we get 2(x^2 + y^2) = 1 + 4 (xy terms cancel out)
Hence x^2 + y^2 = 5/2
Hence Sufficient

Here is my approach

1 and 2 are definitely not sufficient alone.

now from 1 and 2 :-

2xy + 1 = 4 - 2xy
4xy = 3
xy = 3/4

now substituting value of xy in 1 and 2


1 x^2 + y^2 = 2 * 3/4 + 1 = 5/2

2 x^2 + y^2 = 4 - 2 * 3/4 = 4 - 3/2 = 5/2


I guess OA is wrong unless someone has some other method to prove OA is right.

[Blue_Skies]

77. What is the value of x^2 + y^2 ?
(1) x^2 + y^2 = 2xy + 1
(2) x^2 + y^2 = 4 - 2xy

I think the answer is correct.

From 1) (x-y)^2 = 1 So x-y = plus minus 1. Their can be many solutions:
Just for +1 : (1,0)(2,1),(3,2).....(a,a-1)
Similiarly for -1 : their can be infinite solns.

From 2)
(x+y) ^ 2 = plus minus 2.
If you think about it every set of the form (2,0),...(a,a-2) will satisfy this for plus 2. Similarly for -2.

Even if you combine the results of 1 and 2 . You don't get anything because their is no overlap. Hence E is the answer.

[GMATGuruNY]

77. What is the value of x^2 + y^2 ?
(1) x^2 + y^2 = 2xy + 1
(2) x^2 + y^2 = 4 - 2xy

Statement 1:
x² - 2xy + y² = 1.
(x-y)² = 1.
Thus, x-y = 1 or x-y = -1.
No way to determine the value of x² + y².
INSUFFICIENT.

Statement 2:
x² + 2xy + y² = 4.
(x+y)² = 4.
Thus, x+y = 2 or x+y = -2.
No way to determine the value of x² + y².
INSUFFICIENT.

Statements combined:
Adding x² - 2xy + y² = 1 to x² + 2xy + y² = 4, we get:
(x² - 2xy + y²) + (x² + 2xy + y²) = 1+4
2x² + 2y² = 5
x² + y² = 5/2.
SUFFICIENT.

The correct answer is C.

[iamniladri]

I do agree with you but hemant_rajput says OA is E
Thanks

77. What is the value of x^2 + y^2 ?
(1) x^2 + y^2 = 2xy + 1
(2) x^2 + y^2 = 4 - 2xy

Statement 1:
x² - 2xy + y² = 1.
(x-y)² = 1.
Thus, x-y = 1 or x-y = -1.
No way to determine the value of x² + y².
INSUFFICIENT.

Statement 2:
x² + 2xy + y² = 4.
(x+y)² = 4.
Thus, x+y = 2 or x+y = -2.
No way to determine the value of x² + y².
INSUFFICIENT.

Statements combined:
Adding x² - 2xy + y² = 1 to x² + 2xy + y² = 4, we get:
(x² - 2xy + y²) + (x² + 2xy + y²) = 1+4
2x² + 2y² = 5
x² + y² = 5/2.
SUFFICIENT.

The correct answer is C.

[GMATGuruNY]

I do agree with you but hemant_rajput says OA is E
Thanks

The OA is incorrect.
The correct answer is C.

The two statements combined imply the following cases:
Case 1: x-y = 1 and x+y = 2
Here, x=3/2 and y=1/2, with the result that x² + y² = 5/2.

Case 2: x-y = 1 and x+y = -2
Here, x=-1/2 and y=-3/2, with the result that x² + y² = 5/2.

Case 3: x-y = -1 and x+y = 2
Here, x=1/2 and y=3/2, with the result that x² + y² = 5/2.

Case 4: x-y = -1 and x+y = -2
Here, x=-3/2 and y=-1/2, with the result that x² + y² = 5/2.

In each case, x² + y² = 5/2.
SUFFICIENT.

The correct answer is C.