Question

Let $z$ be a complex number such that $|z| + z = 3 + i$ (where i = $\sqrt{-1}$ ). Then $|z|$ is equal to :

• A. $\frac{5}{4}$
• B. $\frac{\sqrt{41}}{4}$
• C. $\frac{\sqrt{34}}{3}$
• D. $\frac{5}{3}$

Answer 1

Answer: (D)

$\frac{5}{3}$

Answer 2

The correct answer is D:$\frac{5}{3}$
Given that;
$|z| + z = 3 + i$(where $i=\sqrt{-1}$)
$z = 3 - |z| + i$
Let $3 - |z| = a$
$\Rightarrow |z| = (3 - a)-(i)$
$\Rightarrow z=a+i$
$\Rightarrow \left|z\right| =\sqrt{a^{2}+1}-(ii)$
Now equate (i) and (ii)
$(3-a)^2=(\sqrt{a^2+i^2})^2$
$\Rightarrow 9+a^{2} -6a =a^{2}+1$
$\Rightarrow a = \frac{8}{6} =\frac{4}{3}$
$\Rightarrow \left|z\right| = 3 - \frac{4}{3} = \frac{5}{3}$