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E?
from stat 1=> a^2 = b^2.....insuff take a=1 b=1 and answer is a=|b| yes
take a=-1 and b=1 and answer is a=|b| no
from stat 2 => b=|a| doesn't mean that a=|b|
combine 1 and 2 (seems to me that the statements are saying nothing substantially different)
hope it's E (I'm a bit tired to combine the statements,maybe it will be better if I go to bed )

from (i), b=lcl, which means b^2=c^2. if so, then
a^2+c^2=2b^2
a^2+b^2=2b^2
a^2 =b^2
a = + or - b, which means a = lbl.............. sufficient.

from (ii), b= lal, which also means b^2=a^2
a^2+c^2=2b^2
b^2+c^2=2b^2
c^2=b^2. then again the same process as in (i)
a^2+c^2=2b^2
a^2+b^2=2b^2
a^2 =b^2
a = + or - b, which means a = lbl.............. sufficient.

Don't be fooled by the seemly complexity of this question.

The question asks if a=|b|. With absolute value questions your first relection should be "do we know it's sign?" |b| is non negative. If we aren't able to determine if a is non negative then we can't determine if a=|b|.

Now look at the stem: a^2-b^2=b^2-c^2
and the two choices:
b=|c| and b=|a|
All we know is b is non negative. Do we know anything about a's sign? No! _________________

Keep on asking, and it will be given you;
keep on seeking, and you will find;
keep on knocking, and it will be opened to you.

Don't be fooled by the seemly complexity of this question.

The question asks if a=|b|. With absolute value questions your first relection should be "do we know it's sign?" |b| is non negative. If we aren't able to determine if a is non negative then we can't determine if a=|b|.

Now look at the stem: a^2-b^2=b^2-c^2 and the two choices: b=|c| and b=|a| All we know is b is non negative. Do we know anything about a's sign? No!

It took me just over 60 secs to choose 'E'. Hong's approach reduces this to a less that 10 sec question. Good one, that!

Think it'll be E
as the first statement only proves that a^2 = b^2
for a to equal |b|, we would have to be sure that a is positive which we can't be...
second statement doesn't give any new information
even with both together there is no way of knowing that a is positive

Don't be fooled by the seemly complexity of this question.

The question asks if a=|b|. With absolute value questions your first relection should be "do we know it's sign?" |b| is non negative. If we aren't able to determine if a is non negative then we can't determine if a=|b|.

Now look at the stem: a^2-b^2=b^2-c^2 and the two choices: b=|c| and b=|a| All we know is b is non negative. Do we know anything about a's sign? No!

Honghu, I am not sure if your approach to this particular question is right, because the question is really asking us whether a = b irrespective of their signs, so this can be deduced to is a^2 = b^2.