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I need help with - Complex numbers and quadratic equations - JEE Main-11

 A  complex  number  z  is  said  to  be   unimodular if  \left | z \right |=1. Suppose z1 and z2 are complex numbers such that

     \frac{z_{1-2z_{2}}}{2-z_{1}\bar{z}_{2}} is unimodular and z2 is not unimodular.Then the point \frac{z_{1-2z_{2}}}{2-z_{1}\bar{z}_{2}}1 lies on a :

  • Option 1)

    straight line parallel to x-axis.

  • Option 2)

    straight line parallel to y-axis.

  • Option 3)

    circle of radius 2.

  • Option 4)

    circle of radius \sqrt{2}

 
Answers (2)
128 Views
N neha

As we have learned

Property of conjugate of complex number -

z\bar{z}=\left |z \right |^{2}

- wherein

  z=x+iy\bar{z}=conjugate \: of\: z   

 \left |z \right |=\sqrt{x^{2}+y^{2}}

 

 Given , \left | \frac{z_1-2z_2}{2-z_1\bar{z}_2} \right |= 1

\Rightarrow |(z_1-2z_2)|^2= |(z-z_1\bar{z}_2)|^2

\Rightarrow (z_1-2z_2)(\bar{z}_1-2\bar{z} _2)= (2-z_1\bar{z}_2)(2-\bar{z}_1z_2)

\Rightarrow (z_1)^2 (1-|z_2|^2)= 4 (1-|z_2|^2)

\Rightarrow |z_1|^2 = 4 (\because |z|\neq 1)

\Rightarrow |z_1|= 2

 

 

 

 

 

 


Option 1)

straight line parallel to x-axis.

Option 2)

straight line parallel to y-axis.

Option 3)

circle of radius 2.

Option 4)

circle of radius \sqrt{2}

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