C.
Statement 1a+b > 0
Let a=b=1 => (1/2 > 0) => Yes
Lets a=0 & b=1 => (1 > 1 ) => No
Thus, the statement is insufficient.
Statement 2b^2 - a^2 > 1
Let a=1 & b=5 => (1/6 > 4) => No
Let a=0 & b=-2 => (1/-2 > -2) => Yes
Thus, statement 2 is also insufficient.
Combining both the statements:
We know that b has to be the bigger value since b^2 > 1 + a^2 ( which is always a +ve value )
Thus, a+b will also have be +ve ( since we cannot have a negative value for b )
Therefore, we can consider the above example where a=1 & b=5 we get a definite NO.
Hence, the answer is C.
Hope this reasoning is correct!
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