Question

If \( z_1, z_2 \) are complex numbers such that \( \dfrac{2z_1}{3z_2} \) is a purely imaginary number, then the value of \[ \left|\frac{z_1 - z_2}{z_1 + z_2}\right| \] is:

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

If ratio is purely imaginary: \begin{itemize} \item Write it as \( it \). \item Substitute and simplify modulus directly. \end{itemize}

Answer 1

The Correct Option is A

Concept: A complex number is purely imaginary if its real part is zero. Given: \[ \frac{2z_1}{3z_2} \text{ is purely imaginary} \Rightarrow \frac{z_1}{z_2} \text{ is purely imaginary} \] So: \[ \frac{z_1}{z_2} = i t, \quad t \in \mathbb{R} \] Thus: \[ z_1 = i t z_2 \] Step 1: {\color{red}Substitute into expression.} \[ \frac{z_1 - z_2}{z_1 + z_2} = \frac{it z_2 - z_2}{it z_2 + z_2} = \frac{it - 1}{it + 1} \] Step 2: {\color{red}Find modulus.} \[ \left|\frac{it - 1}{it + 1}\right| = \frac{|it - 1|}{|it + 1|} \] But: \[ |it - 1| = \sqrt{1 + t^2} \] \[ |it + 1| = \sqrt{1 + t^2} \] So ratio = 1.