Z 1 and z 2 are complex numbers such that z 1-2 z 2-2- z 1z2 is unimodular and Z 2 is not unimodular- Find - z 1-A- 2B- 1C- 4D- none of these

The correct option is A 2| z_1 2z_2/2 z_1 z̅_̅2̅ |=1 ⇒ |z_1 2 z_2|=|2 z_1 z̅_̅2̅| ⇒ (z_1 2z_2)(z̅_̅1̅ ̅2̅z̅̅̅_̅̅̅2̅̅̅)=(2 z_1 z̅_̅2̅)(2 z̅_̅1̅ z_2) ⇒ |z_1|^2 | ...

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Byju's Answer

Standard XII

Mathematics

Properties of Argument

z 1 and z 2 a...

Question

$z_{1}$ and $z_{2}$ are complex numbers such that $\frac{z_{1} - 2 z_{2}}{2 - z_{1}_{2}}$ is unimodular and Z2 is not unimodular. Find $| z_{1} |$

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none of these

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Solution

The correct option is B 2
$∣ ∣ \frac{z_{1} - 2 z_{2}}{2 - z_{1}_{2}} ∣ ∣ = 1 \Rightarrow | z_{1} - 2 z_{2} | = | 2 - z_{1}_{2} |$
$\Rightarrow (z_{1} - 2 z_{2}) (¯ z_{1} - 2_{2}) = (2 - z_{1}_{2}) (2 -_{1} z_{2}) \Rightarrow | z_{1} |^{2} - | z_{1} |^{2} | z_{2} |^{2} + 4 | z_{2} |^{2} - 4 = 0 \Rightarrow | z_{1} | = 2$

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z1 and z2 are complex numbers such that z1−2z22−z1¯z2 is unimodular and Z2 is not unimodular. Find |z1|

The correct option is B 2∣∣z1−2z22−z1¯z2∣∣=1⇒|z1−2z2|=|2−z1¯z2| ⇒(z1−2z2)(¯z1−2¯z2)=(2−z1¯z2)(2−¯z1z2)⇒|z1|2−|z1|2|z2|2+4|z2|2−4=0⇒|z1|=2

$z_{1}$ and $z_{2}$ are complex numbers such that $\frac{z_{1} - 2 z_{2}}{2 - z_{1}_{2}}$ is unimodular and Z2 is not unimodular. Find $| z_{1} |$