Mathematics - Complex Number Question with Solution | TestHub

$|z+2|=1,\mathrm{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$
Let $z+2=\cos \theta +i\sin \theta$ $\begin{array}{rl}& \frac{1}{z+2}=\cos \theta -i\sin \theta \\ & \Rightarrow \frac{z+1}{z+2}=1-\frac{1}{z+2}=1-(\cos \theta -i\sin \theta )\\ & =(1-\cos \theta )+\mathrm{isin}\theta \\ & \mathrm{Im}\left(\frac{z+1}{z+2}\right)=\sin \theta ,\sin \theta =\frac{1}{5}\\ & \cos \theta =\pm \sqrt{1-\frac{1}{25}}=\pm \frac{2\sqrt{6}}{5}\\ & |\mathrm{Re}(\overline{z+2})|=\frac{2\sqrt{6}}{5}\end{array}$