If |Z|= 1 , then the locus of Z₁ which moves such that${\mathrm{Z}}_1=\frac{5-4\mathrm{Z}}{4\mathrm{Z}-2}$in the complex plane, is

$z_1=\frac{5-4z}{4z-2}$

$\Rightarrow \mathrm{z}=\frac{5+2{\mathrm{z}}_1}{4\left({\mathrm{z}}_1+1\right)}\left|\mathrm{z}\right|=1\Rightarrow \left|\frac{5+2{\mathrm{z}}_1}{4\left({\mathrm{z}}_1+1\right)}\right|=1$

$\begin{array}{l} \\ \Rightarrow \left|{\mathrm{z}}_1+\frac{5}{2}\right|=2\left|{\mathrm{z}}_1+1\right|\\ \Rightarrow {\left(\mathrm{x}+\frac{5}{2}\right)}^2+{\mathrm{y}}^2=4\left((\mathrm{x}+1{)}^2+{\mathrm{y}}^2\right)\\ {\mathrm{x}}^2+{\mathrm{y}}^2+\mathrm{x}-\frac{3}{4}=0\Rightarrow \text{ centre }=\left(-\frac{1}{2},0\right)\text{ radius }=1\end{array}$

$\text{ circle passes through all 4 quadrants}$