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Which of the following are true?

Source:

March 22, 2006
complex numbers

Problem Statement

Consider the statements:
(I)  a2+b2=0\text{(I)}~~\sqrt{a^2+b^2}=0 (II)  a2+b2=ab\text{(II)}~~\sqrt{a^2+b^2}=ab (III)  a2+b2=a+b\text{(III)}~~\sqrt{a^2+b^2}=a+b (IV)  a2+b2=a−b\text{(IV)}~~\sqrt{a^2+b^2}=a-b,
where we allow aa and bb to be real or complex numbers. Those statements for which there exist solutions other than a=0a=0 and b=0b=0 are:
(A) (I),(II),(III),(IV)(B) (II),(III),(IV)(C) (I),(III),(IV)(D) (III),(IV)(E) (I)\text{(A)} \ \text{(I)},\text{(II)},\text{(III)},\text{(IV)} \qquad \text{(B)} \ \text{(II)},\text{(III)},\text{(IV)} \qquad \text{(C)} \ \text{(I)},\text{(III)},\text{(IV)} \qquad \text{(D)} \ \text{(III)},\text{(IV)} \qquad \text{(E)} \ \text{(I)}
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