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$

A. z = \bar w \\

B. z\bar w = 1 \\

C. z\bar zw = 2 \\

D. z\bar w = 4 \\

$

Answer
Verified

In this question, two complex numbers are given, and we need to determine the relation between the conjugate of one complex number with the original form of another complex number for which we need to follow the concept of argument and properties of the complex conjugate of the numbers.

\[

|z{|^2}w - |w{|^2}z = z - w{\text{ }} \\

|z{|^2}w + w = z + |w{|^2}z \\

w\left( {|z{|^2} + 1} \right) = z\left( {|w{|^2} + 1} \right) \\

\dfrac{w}{z} = \left( {\dfrac{{|w{|^2} + 1}}{{|z{|^2} + 1}}} \right) - - - - (i) \\

\]

As equation (i), does not have any imaginary part so, from equation (i), it can also be written as:

\[\dfrac{{\bar w}}{{\bar z}} = \left( {\dfrac{{|w{|^2} + 1}}{{|z{|^2} + 1}}} \right) - - - - (ii)\]

Dividing equation (i) and equation (ii) we get,

\[

\dfrac{{\left( {\dfrac{w}{z}} \right)}}{{\left( {\dfrac{{\bar w}}{{\bar z}}} \right)}} = \dfrac{{\left( {\dfrac{{|w{|^2} + 1}}{{|z{|^2} + 1}}} \right)}}{{\left( {\dfrac{{|w{|^2} + 1}}{{|z{|^2} + 1}}} \right)}} \\

\dfrac{{w\bar z}}{{\bar wz}} = 1 \\

w\bar z = \bar wz = 1 \\

\]

Hence, for two complex numbers $z{\text{ and }}w$, such that $|z{|^2}w - |w{|^2}z = z - w{\text{ and, (}}z \ne w{\text{)}}$ then $z\bar w = 1$.

Option B is correct.

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