Question

If $z$ is a complex number such that $z+\left|z\right|=8+12i$ then the value of $\left|z^2\right|$ is equal to

• A. 228
• B. 144
• C. 121
• D. 169
• E. 189

Answer 1

Answer: (D)

Let $z=x+iy$
Then, we have $z+|z|=8+12i$
$\Rightarrow (x+iy)+|x+iy|=8+12i$
$\Rightarrow \left(x+\sqrt{x^2+y^2}\right)+iy=8+12i$
On comparing the real and imaginary part, we get
and $y=12$
and $x+\sqrt{x^2+y^2}=8$
$\Rightarrow \sqrt{x^2+144}=8-x$
On squaring both sides, we get
$x^2+144=64+x^2-16x$
$\Rightarrow 16x=-80$
$\Rightarrow x=-5$
$\therefore z=x+iy=-5+i\cdot 12$
Then, $|z|=\sqrt{25+144}=\sqrt{169}=13$
$\Rightarrow |z{|}^2=169$
$\Rightarrow \left|z^2\right|=169$
$\left(\because \left|z^n\right|=|z{|}^n\right)$