Question

If $z$ is a complex number such that $z+|z|=8+12i$, then the value of $|z^2|$ is

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

Answer 1

The correct option is D $169$

Let $z=x+iy$
Then, we have $z+|z|=8+12i$
$\Rightarrow (x+iy)+|x+iy|=8+12l$
$\Rightarrow (x+\sqrt{x^2+y^2})+iy=8+12i$
On comparing the real and imaginary part, we get
$y=12$
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, $|z|=\sqrt{25+144}=\sqrt{169}=13$
$\Rightarrow |z{|}^2=169$
$\Rightarrow |z^2|=169(\because |z^n|=|z{|}^n)$.