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: Understanding the Question:
We need to find the modulus of a complex number $z$ that satisfies a given algebraic equation.

Step 2: Key Formula or Approach:
Let $z = x + iy$. Then $|z| = \sqrt{x^2 + y^2}$. Compare real and imaginary parts.

Step 3: Detailed Explanation:
Given: $\sqrt{x^2 + y^2} + (x + iy) = 3 + i$.
Equating imaginary parts: $y = 1$.
Equating real parts: $\sqrt{x^2 + 1} + x = 3$.
\[ \sqrt{x^2 + 1} = 3 - x \] Squaring both sides: \[ x^2 + 1 = (3 - x)^2 \] \[ x^2 + 1 = 9 - 6x + x^2 \] \[ 6x = 8 \Rightarrow x = \frac{4}{3} \] Modulus $|z| = \sqrt{x^2 + y^2}$: \[ |z| = \sqrt{(\frac{4}{3})^2 + 1^2} = \sqrt{\frac{16}{9} + 1} = \sqrt{\frac{25}{9}} = \frac{5}{3} \]

Step 4: Final Answer:
The value of $|z|$ is $5/3$.