Let $z$ be a complex number satisfying $|z|^{3}+2 z^{2}+4 \bar{z}-8=0$, where $\bar{z}$ denotes the

Question

Let $z$ be a complex number satisfying $|z|^{3}+2 z^{2}+4 \bar{z}-8=0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero. Match each entry in List-I to the correct entries in List-II. The correct option is

Professor PC Thomas Classes · Answer

(b) : Let $Z=r(\cos 	heta+i \sin 	heta)$\ $r^{3}+2 r^{2} e^{2 i 	heta}+4 r e^{-i 	heta}-8=0$\ Comparing real and imaginary part $$\begin{equation*}r^{3}+2 r^{2} \cos 2 	heta+4 r \cos 	heta=8 	ag{i} \end{equation*}$$ and $4 r^{2} \sin 	heta \cos 	heta=4 r \sin 	heta$\ $\Rightarrow \quad r \cos 	heta=1$ put in (i) $$\begin{equation*}\Rightarrow \cos 2 	heta=2 \cos ^{2} 	heta-1=\frac{2-r^{2}}{r^{2}} 	ag{ii} \end{equation*}$$ $\Rightarrow \quad r^{3}+2\left(2-r^{2}ight)=4 \Rightarrow r=2$\ (P) $|Z|^{2}=4$\ (Q) $|Z-\bar{Z}|^{2}=|2 i r \sin 	heta|^{2}$\ $=4 r^{2}\left(1-\cos ^{2} 	hetaight)=16-4=12$\ (R) $|Z|^{2}+|Z+\bar{Z}|^{2}=4+4=8$\ (S) $|Z+1|^{2}=(Z+1)(\bar{Z}+1)=Z \bar{Z}+Z+\bar{Z}+1$ $=4+2+1=7$\