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Question

Let Z1 and Z2 be two complex numbers such that Z12Z22z1¯¯¯Z2=1 and |Z2|1, find |Z1|

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Solution

Given:Z12Z22Z1¯Z2=1
To find: |Z1|
On squaring we get
|Z12Z2|2=|2Z1¯Z2|2

Now we know that |Z|2=Z¯Z
(Z12Z2)(¯Z12¯Z2)=(2Z1¯Z2)(2¯Z1Z2)

|Z1|22Z1¯Z22Z2¯Z1+4|Z2|2=42¯Z1Z22Z1¯Z2+|Z1|2|Z2|2

|Z1|2|Z1|2|Z2|2=44|Z2|2

|Z1|2{1|Z2|2}=4{1|Z2|2}

Since |Z2|1

|Z1|2=4

|Z1|=2

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