Does x + y = 0? (1) xy < 0 (2) x^2 = y^2

1) xy<0, it means that x and y has different signs, but not necessarily equal. NS
2) (x-y)(x+y)=0
either x=y or x=-y, as we are not certain that x+y=0, then it's NS
BOTH
as xy<0, the signs of x and y are different, then x=-y, or x+y=0 Suf

Does x + y = 0? (OR is -x=y OR x=-y)

(1) xy<0
(2) x^2 = y^2

So
1: xy<0 so either x OR y <0 Not sufficient

2. x^2 =y^2 . |x| = |y|. But we dont know anything about the sign of either x OR y. Not sufficient

Both together
xy<0 and x^2 =y^2 : So either one of them is negative, and they have same aboslute value. Suffient to answer the question x+y=0

Statement 1
xy < 0
x can be +ive or -ive, or can be = to y
y can be +ive or -ive, or can be = to x

With those cases in mind, inline are some samples
x=1, y=-1, will answer the question as Yes
x=2, y =-1, will answer the question as No

Statement 2
x can be +ive or -ive
y can be +ive or -ive

With those cases in mind, inline are some samples
x=1, y=-1, will answer the question as Yes
x=1, y =1, will answer the question as No

Combine both the statements and you will always get a Yes
If x is +ive, y has to be -ive
If y is +ive, x has to be -ive.

With those cases in mind, inline are some samples
x=1, y=-1, will answer the question as Yes
x=-1, y =1, will answer the question as Yes

Correct answer C.

Please correct me if I am wrong but wont the answer be E?
This is because it is not mentioned that x and y are integers. So in case of both the statements combined, the value of x and y could be a decimal for eg., 0.1 which would satisfy both the conditions, but x+y wont be equal to zero

Think about it, if x = 0.1, what would be the value of y, and what would be the value of x + y?

Oh right! Got it now! Thank you so much!