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 $y=7$
$\Rightarrow \sqrt{x^{2} + y^{2}}=1-x\Rightarrow x^{2}+y^{2}=1+x^{2}-2x$
$y^{2}=1-2x$
$49-1=-2x$
$x=-24$
$\left|z\right|^{2}=x^{2}+y^{2}=576+49=625$
$\left|z\right|=25$
$\sqrt{\left(\left|z + \bar{z}\right|\right)^{2} + \left(\left|z - \bar{z}\right|\right)^{2}}=\sqrt{\left(1^{2} + 1^{2}\right) \left(\left(\left|z\right|\right)^{2} + \left(\left|\bar{z}\right|\right)^{2}\right)}=\sqrt{2 \times 2 \times 625}=\sqrt{4 \times 625}=2\times 25=50$
$\sqrt{\left|z + \bar{z}\right|^{2} + \left|z - \bar{z}\right|^{2}}=50$