Let \(w\ (\mathrm{Im}\,w\ne 0)\) be a complex number. Then, the set of all complex numbers \(z\) satisfying the equation \(w-\overline{w}z=k(1-z)\), for some real number \(k\), is
Show Hint
- \(\{z: z\ne 1\}\)
- \(\{z: |z|=1,\ z\ne 1\}\)
- \(\{z: z=\overline{z}\}\)
- \(\{z: |z|=1\}\)
The Correct Option is B
Solution and Explanation
So \(z=\frac{w-k}{\overline w-k}\). As \(k\) is real, \(\overline w-k=\overline{w-k}\), hence \(|z|=\frac{|w-k|}{|\overline{w-k}|}=1\).
Also \(z=1\) would need \(w-k=\overline w-k\), i.e. \(w=\overline w\), impossible because \(\mathrm{Im}\,w\ne 0\). So \(|z|=1,\ z\ne1\).
Answer: (2) \(\{z: |z|=1,\ z\ne 1\}\)
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