Question

Let $z_1,z_2$ be two complex numbers such that $z_1+z_2$ and
$z_1{z}_2$ both are real , then

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Lets $z_1=a+ib,z_2=c+id,$ then
$z_1+z_2$ is real $\Rightarrow (a+b)+i(b+d)$ is real
$\Rightarrow b+d=0\Rightarrow d=-b..............\left(i\right)$
$z_1{z}_2$ is real $\Rightarrow (ad-bd)+i(ac+bc)$ is real
$\Rightarrow ad+dc=0\Rightarrow a(-b)+bc=0\Rightarrow a=c$
$\because z_1=a+ib=c-id=\left(\overline{z_2}\right(\because a=candb=-d)$