Question

Let $z$ and $w$ be two complex numbers such that $w=z\overline{z}-2z+2$, $\left|\frac{(z+i)}{(z–3i)}\right|=1$ and $Re\left(w\right)$ has a minimum value. Then, the minimum value of $n\in N,$ for which $w^n$ is real, is equal to ____

Answer 1

Step 1: Solving $\left|\frac{(z+i)}{(z–3i)}\right|=1$

Considering $z=x+iy$

Given that,

$\left|\frac{(z+i)}{(z–3i)}\right|=1$

$$\begin{gathered} \Rightarrow |z+i|=|z–3i| \\ \Rightarrow \left|x+\left(y+1\right)i\right|=\left|x+\left(y-3\right)i\right| \\ \Rightarrow \sqrt{x^2+{\left(y+1\right)}^2}=\sqrt{x^2+{\left(y-3\right)}^2} \\ \Rightarrow x^2+{\left(y+1\right)}^2=x^2+{\left(y-3\right)}^2\text{[squaring both sides]} \\ \Rightarrow y^2+1+2y=y^2+9-6y \\ \Rightarrow y=1 \end{gathered}$$

Step 2: Evaluating expression for $w$

Given that $w=z\overline{z}-2z+2$

$$\begin{gathered} \Rightarrow w=(x+iy)(x-iy)-2(x+iy)+2[\because \overline{(x+iy)}=(x-iy)] \\ \Rightarrow w=x^2+y^2–2x–2iy+2 \\ \Rightarrow w=(x^2–2x+3)–2i\left[\because y=1\right] \end{gathered}$$

$$\begin{gathered} \therefore Re\left(w\right)=x^2–2x+3 \\ \Rightarrow Re\left(w\right)=\left(x^2–2x+1\right)+2 \\ \Rightarrow Re\left(w\right)={(x–1)}^2+2 \\ \therefore Re\left(w\right)minatx=1 \end{gathered}$$

Step 3: Finding value of $n\in N,$ for which $w^n$ is real

From minimum value of $x=1\&y=1$ we get

$z=1+i$

Substituting this value of $z$ into $w=z\overline{z}-2z+2$

$$\begin{gathered} \Rightarrow w=(1+i)(1-i)-2(1+i)+2 \\ \Rightarrow w=1+1–2–2i+2 \\ \Rightarrow w=2(1–i) \\ =\frac{2\sqrt{2}(1–i)}{\sqrt{2}}[\text{multiplying and dividing by}\sqrt{2}] \\ =2\sqrt{2}\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}i\right) \\ =2\sqrt{2}\left(\cos \frac{\mathrm{\pi }}{4}-i\sin \frac{\mathrm{\pi }}{4}\right)\left[\because \cos \frac{\mathrm{\pi }}{4}=\sin \frac{\mathrm{\pi }}{4}=1\right] \\ \Rightarrow w=2\sqrt{2}{e}^{-i\left(\frac{\mathrm{\pi }}{4}\right)} \\ \Rightarrow w^4={\left(2\sqrt{2}{e}^{-i\left(\frac{\mathrm{\pi }}{4}\right)}\right)}^4 \\ \Rightarrow w^4=16\times 4e^{-i\mathrm{\pi }} \\ \Rightarrow w^4=-64\left[\because e^{-i\mathrm{\pi }}=-1\right] \end{gathered}$$

Hence, the minimum value of $n\in N,$ for which $w^n$ is real, is equal to $4$