Solve the following

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 z1z2 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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