Question

If \(z\) be a complex number such that \( |z| + z = 2 + i \), then find the value of \( |z| \).

• A. \( \frac{1}{2} \)
• B. \( \frac{3}{4} \)
• C. \( \frac{5}{4} \)
• D. \( 1 \)

Hint:

For equations involving complex numbers, compare real and imaginary parts separately.

Answer 1

The Correct Option is C

Concept: A complex number \(z\) is typically represented in the form \(z = x + iy\), where \(x\) and \(y\) are real numbers representing the real and imaginary parts respectively, and \(i = \sqrt{-1}\). The modulus (or absolute value) of a complex number, denoted by \(|z|\), represents its distance from the origin in the complex plane and is calculated as: \[ |z| = \sqrt{x^2 + y^2} \] When solving equations involving complex numbers, a fundamental principle is that two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

Step 1: Substituting the standard form of a complex number.}
Let the complex number be \[ z = x + iy \] Then, \[ |z| = \sqrt{x^2 + y^2} \] Substituting into the given equation: \[ \sqrt{x^2 + y^2} + (x + iy) = 2 + i \]

Step 2: Equating the imaginary parts.
\[ y = 1 \]

Step 3: Equating the real parts and solving for \(x\).
\[ \sqrt{x^2 + 1} + x = 2 \] \[ \sqrt{x^2 + 1} = 2 - x \] Squaring both sides: \[ x^2 + 1 = 4 + x^2 - 4x \] \[ 4x = 3 \] \[ x = \frac{3}{4} \]

Step 4: Calculating the value of \(|z|\).
\[ |z| = \sqrt{\left(\frac{3}{4}\right)^2 + 1} \] \[ |z| = \sqrt{\frac{9}{16} + \frac{16}{16}} \] \[ |z| = \sqrt{\frac{25}{16}} \] \[ |z| = \frac{5}{4} \]