Question

Let \(z\) and \(w\) be two distinct non-zero complex numbers. If \[ |z|^2w-|w|^2z=z-w, \] then

• A. \(w=\overline{z}^{\,2}\)
• B. \(zw=2\)
• C. \(z\overline{w}=1\)
• D. \(w=\overline{z}\)

Hint:

For complex numbers, use \(|z|^2=z\overline{z}\). This often helps in converting modulus equations into equations involving conjugates.

Answer 1

The Correct Option is C

Step 1: Rearrange the equation.
Given \(|z|^2 w - |w|^2 z = z - w\). Rewrite as \(|z|^2 w - z = |w|^2 z - w\), i.e., \(z(|w|^2 - 1) = w(|z|^2 - 1)\), so \(\tfrac{z}{w} = \tfrac{|z|^2-1}{|w|^2-1}\) (if \(|w|^2 \neq 1\)).

Step 2: Test \(z\bar{w} = 1\).
From the original: group as \(w(|z|^2 - 1) = z(|w|^2 - 1)\). If \(z\bar{w}=1\), then \(|z|^2 = z\bar{z}\) and \(\bar{w} = 1/z\), so \(|w|^2 = w/z\). Substituting verifies the equation holds. Since \(z\) and \(w\) are distinct and non-zero, this is the only consistent relation.
\[\boxed{z\bar{w} = 1}\]