JEE Mains · Maths · STD 11 - 4.1 complex nubers
Let \(z\) and \(w\) be two complex numbers such that \(w=z \bar{z}-2 z+2,\left|\frac{z+i}{z-3 i}\right|=1\) and \(\operatorname{Re}(w)\) has minimum value. Then, the minimum value of \(n \in N\) for which \(w ^{ n }\) is real, is equal to..........
- A \(5\)
- B \(2\)
- C \(4\)
- D \(6\)
Answer & Solution
Correct Answer
(C) \(4\)
Step-by-step Solution
Detailed explanation
\(\omega=z \bar{z}-2 z+2\) \(\left|\frac{z+i}{z-3 i}\right|=1\) \(\Rightarrow \quad|z+i|=|z-3 i|\) \(\Rightarrow \quad z=x+i, \quad x \in R\) \(\omega=(x+i)(x-i)-2(x+i)+2\) \(=x^{2}+1-2 x-2 i+2\) \(\operatorname{Re}(\omega)=x^{2}-2 x+3\) For…
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