Question

The complex number $z$ satisfying $z+\left|z\right|=1+7i,$ then the value of $\sqrt{{\left|z+\bar{z}\right|}^2+{\left|z-\bar{z}\right|}^2}$ is

• A. $30$
• B. $40$
• C. $50$
• D. None of those

Answer 1

Answer: (C)

Let $z=x+iy$
$x+iy+\sqrt{x^2+y^2}=1+7i$
On comparing real and imaginary parts,
$x+\sqrt{x^2+y^2}=1$ and $y=7$
$\Rightarrow \sqrt{x^2+y^2}=1-x\Rightarrow x^2+y^2=1+x^2-2x$
$y^2=1-2x$
$49-1=-2x$
$x=-24$
${\left|z\right|}^2=x^2+y^2=576+49=625$
$\left|z\right|=25$
$\sqrt{{\left(\left|z+\bar{z}\right|\right)}^2+{\left(\left|z-\bar{z}\right|\right)}^2}=\sqrt{\left(1^2+1^2\right)\left({\left(\left|z\right|\right)}^2+{\left(\left|\bar{z}\right|\right)}^2\right)}=\sqrt{2\times 2\times 625}=\sqrt{4\times 625}=2\times 25=50$
$\sqrt{{\left|z+\bar{z}\right|}^2+{\left|z-\bar{z}\right|}^2}=50$