Q76 / 155 · WBJEE 2025 · Mathematics

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

  • Correct answer
    A
    $1$
  • B
    $2$
  • C
    $3$
  • D
    $4$

Solution

Given: \(\frac{2 z_1}{3 z_2}\) is purely imaginary \(\Rightarrow \operatorname{Re}\left(\frac{z_1}{z_2}\right)=0 \Rightarrow \frac{z_1}{z_2}\) is purely imaginary Let \(\frac{z_1}{z_2}=i k\) for some real \(k\) Then \(z_1=i k z_2\) Now compute: \(\left|\frac{z_1-z_2}{z_1+z_2}\right|=\left|\frac{i k z_2-z_2}{i k z_2+z_2}\right|=\left|\frac{z_2(i k-1)}{z_2(i k+1)}\right|=\left|\frac{i k-1}{i k+1}\right|\) Let \(z=\frac{i k-1}{i k+1}\) Now, \(|z|=\left|\frac{i k-1}{i k+1}\right|=\sqrt{\frac{\left(k^2+1\right)}{\left(k^2+1\right)}}=1\)

Hence, the correct answer is option: 'A'.

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