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

The Correct Option is D

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)
\((3-a)^2=(\sqrt{a^2+i^2})^2\)
\(\Rightarrow 9+a^{2} -6a =a^{2}+1\)
\(\Rightarrow a = \frac{8}{6} =\frac{4}{3}\)
\(\Rightarrow \left|z\right| = 3 - \frac{4}{3} = \frac{5}{3}\)