• # question_answer  Let ${{z}_{1}},{{z}_{2}}$ be two complex numbers such that ${{z}_{1}}+{{z}_{2}}$ and ${{z}_{1}}{{z}_{2}}$ both are real, then [RPET 1996] A) ${{z}_{1}}=-{{z}_{2}}$ B) ${{z}_{1}}={{\bar{z}}_{2}}$ C) ${{z}_{1}}=-{{\bar{z}}_{2}}$ D) ${{z}_{1}}={{z}_{2}}$

Let ${{z}_{1}}=a+ib,{{z}_{2}}=c+id$, then  ${{z}_{1}}+{{z}_{2}}$ is real    Þ $(a+c)+i(b+d)$is real Þ $b+d=0$     Þ $d=-b$            .....(i) ${{z}_{1}}{{z}_{2}}$ is real        Þ $(ad-bd)+i(ac+bc)$is real Þ  $ad+bc=0$ Þ $a(-b)+bc=0$Þ $a=c$ \${{z}_{1}}=a+ib=c-id={{\bar{z}}_{2}}$ $(\because a=c$and $b=-d)$