Question

Let $z$ be a complex number such that $|z|+z=3+i$ where $i=\sqrt{-1}$, then $|z|=$

• A. $5/3$
• B. $3/5$
• C. $4/3$
• D. $5/4$

Hint:

Always separate real and imaginary parts when dealing with equations involving complex numbers and their moduli.

Answer 1

The Correct Option is A

Step 1: Concept
Let $z = x + iy$, so $|z| = \sqrt{x^2+y^2}$.

Step 2: Analysis
$\sqrt{x^2+y^2} + (x + iy) = 3 + i$.
Equate real and imaginary parts:
Imaginary: $y = 1$.
Real: $\sqrt{x^2+1} + x = 3$.

Step 3: Calculation
$\sqrt{x^2+1} = 3 - x$
$x^2 + 1 = 9 + x^2 - 6x$
$6x = 8 \implies x = 4/3$.
$|z| = \sqrt{(4/3)^2 + 1^2} = \sqrt{16/9 + 1} = \sqrt{25/9} = 5/3$.

Step 4: Conclusion
Hence, $|z| = 5/3$. Final Answer: (A)