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Algebra Difficulty 2.1 Junior Find the answer

Let Z1=a+biZ_1 = a + bi and Z2=c+diZ_2 = c + di, then the necessary and sufficient condition for the complex number Z1⋅Z2Z_1 \cdot Z_2 to be a real number is

Pick one

Solution

Given Z1=a+biZ_1 = a + bi and Z2=c+diZ_2 = c + di,
the product of the complex numbers Z1⋅Z2=(a+bi)(c+di)Z_1 \cdot Z_2 = (a + bi)(c + di)
=(ac−bd)+(ad+bc)i= (ac - bd) + (ad + bc)i
Since Z1⋅Z2Z_1 \cdot Z_2 is a real number,
it follows that ad+bc=0ad + bc = 0,
therefore, the correct choice is A\boxed{\text{A}}.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.