A complex number z is said to be unimodular if |z|=1. Suppose z1 and z2 are complex numbers such that z1−2z22−z1z¯2 is unimodular and z2 is non-unimodular. Then the point z1 lies on a

|z|=1;z2≠1 Given z1−2z22−z1z2=10∴z1−2z22=2−z1z¯22⇒z12+4z22−2z¯1z2−2z1z¯2=4+z12−2z¯1z2−2z1z¯2⇒z22−1z12−4=0∴z2≠1∴z1=2

A complex number z is said to be unimodular if |z|=1. Suppose z1 and z2 are complex numbers such that z1−2z22−z1z¯2 is unimodular and z2 is non-unimodular. Then the point z1 lies on a

|z|=1;z2≠1 Given z1−2z22−z1z2=10∴z1−2z22=2−z1z¯22⇒z12+4z22−2z¯1z2−2z1z¯2=4+z12−2z¯1z2−2z1z¯2⇒z22−1z12−4=0∴z2≠1∴z1=2

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A complex number z is said to be unimodular if $| z | = 1.$ Suppose $z_{1} a n d z_{2}$ are complex numbers such that $\frac{z_{1} - 2 z_{2}}{2 - z_{1} {\bar{z}}_{2}}$ is unimodular and $z_{2}$ is non-unimodular. Then the point $z_{1}$ lies on a

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Straight line parallel to x-axis

Straight line parallel to y-axis

Circle of radius 2

Circle of radius $\sqrt{2}$

answer is C.

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Detailed Solution

$| z | = 1; |z_{2}| \neq 1$

$Given |\frac{z_{1} - 2 z_{2}}{2 - z_{1} z_{2}}| = 10$

$∴ {|z_{1} - 2 z_{2}|}^{2} = {|2 - z_{1} {\bar{z}}_{2}|}^{2}$

$\Rightarrow {|z_{1}|}^{2} + 4 {|z_{2}|}^{2} - 2 {\bar{z}}_{1} z_{2} - 2 z_{1} {\bar{z}}_{2} = 4 + {|z_{1}|}^{2} - 2 {\bar{z}}_{1} z_{2} - 2 z_{1} {\bar{z}}_{2}$

$\Rightarrow ({|z_{2}|}^{2} - 1) ({|z_{1}|}^{2} - 4) = 0$

$∴ |z_{2}| \neq 1$

$∴ |z_{1}| = 2$

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A complex number z is said to be unimodular if |z|=1. Suppose z1 and z2 are complex numbers such that z1−2z22−z1z¯2 is unimodular and z2 is non-unimodular. Then the point z1 lies on a

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A complex number z is said to be unimodular if $| z | = 1.$ Suppose $z_{1} a n d z_{2}$ are complex numbers such that $\frac{z_{1} - 2 z_{2}}{2 - z_{1} {\bar{z}}_{2}}$ is unimodular and $z_{2}$ is non-unimodular. Then the point $z_{1}$ lies on a