Question:
If z1 and z2 are two complex numbers such that z1 + z2 is a real number, then z2 = ____________.
Solution:
Given for two complex numbers, z1 and z2, we have z1+ z2 is real number
Let $z_{1}=x_{1}+i y_{1}$
$z_{2}=x_{2}+i y_{2}$
$\Rightarrow z_{1}+z_{2}=x_{1}+i y_{1}+x_{2}+i y_{2}$
i. e $z_{1}+z_{2}=\left(x_{1}+x_{2}\right)+i\left(y_{1}+y_{2}\right)$
Since $z_{1}+z_{2}$ is real
$\Rightarrow y_{1}+y_{2}=0$
i. e $y_{1}=-y_{2}$
i.e $z_{2}=x_{2}+i y_{2}=x_{2}-i y_{1}$
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