If z_1,z_2 are two distinct complex number such that |z_1-2z_2(1)/(2)-z_1z_2|=2, then

Question

If ${z}_{1},{z}_{2}$ are two distinct complex number such that $|\frac{{z}_{1}-2{z}_{2}}{\frac{1}{2}-{z}_{1}{\vec{z}}_{2}}|=2$, then

Accepted Answer

$\begin{gathered} |\frac{{z}_{1}-2{z}_{2}}{\frac{1}{2}-{z}_{1}{\vec{z}}_{2}}|=2 \ |{z}_{1}-2{z}_{2}|=|1-2{z}_{1}{\vec{z}}_{2}| \ \Rightarrow ({z}_{1}-2{z}_{2})({\vec{z}}_{1}-2{\vec{z}}_{2})=(1-2{z}_{1}{\vec{z}}_{2})(1-2{\vec{z}}_{1}{z}_{2}) \ \Rightarrow {|{z}_{1}|}^{2}+4{|{z}_{2}|}^{2}-2{\vec{z}}_{1}{z}_{2}-2{\vec{z}}_{2}{z}_{1} \ \, =1+4{|{z}_{1}|}^{2}{|{z}_{2}|}^{2}-2{z}_{1}{\vec{z}}_{2}-2{\vec{z}}_{1}{z}_{2} \ \Rightarrow {|{z}_{1}|}^{2}+4{|{z}_{2}|}^{2}-4{|{z}_{1}|}^{2}{|{z}_{2}|}^{2}-1=0 \ \Rightarrow ({|{z}_{1}|}^{2}-1)(1-4{|{z}_{2}|}^{2})=0 \ \Rightarrow |{z}_{1}|=1|{z}_{2}|=\frac{1}{2} \end{gathered}$

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