Question

Let $z$ be a complex number such that $|z|=z+32-24i$. Then $Re\left(z\right)+Im\left(z\right)$ is equal to

• A. $17$
• B. $31$
• C. $25$
• D. $-17$

Answer 1

The correct option is A $17$

$|z|=z+32-24i\Rightarrow z=(|z|-32)+24i\Rightarrow |z{|}^2=(|z|-32{)}^2+{24}^2\Rightarrow |z{|}^2=|z{|}^2-64|z|+{32}^2+{24}^2\Rightarrow 64|z|=8^2(4^2+3^2)\Rightarrow |z|=25$
So,
$25=z+32-24i\Rightarrow z=-7+24i\therefore Re\left(z\right)+Im\left(z\right)=-7+24=17$