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  • Help me understand! - Complex numbers and quadratic equations - JEE Main-15

Let z & w are complex numbers satisfying w= \frac{z}{z-\bar{z}}  then Re\left ( w \right ) equals \left ( it\; is\; given\; z\neq \bar{z} \right )

  • Option 1)

    \frac{-1}{2}

  • Option 2)

    0

  • Option 3)

    1

  • Option 4)

    \frac{1}{2}

 

Answers (1)

\because Re\left ( w \right )= \frac{w+\bar{w}}{2}

\because w=\frac{z}{z-\bar{z}}

so  \because \bar w=\overline{(\frac{z}{z-\bar{z}})}= \frac{\bar{z}}{\bar{z}-\bar{\bar{z}}}= \frac{\bar{z}}{\bar{z}-z}

w+\bar{w}= \frac{z}{z-\bar{z}}+\frac{\bar{z}}{\bar{z}-z}= \frac{z-\bar{z}}{z-\bar{z}}=1

Re\left ( w \right )= \frac{1}{2}

 

Properties of Conjugate of Complex Number -

If w=f(z), then \bar{w}=f(\bar{z})

- wherein

w is a function of complex number z.

 

 


Option 1)

\frac{-1}{2}

This is incorrect

Option 2)

0

This is incorrect

Option 3)

1

This is incorrect

Option 4)

\frac{1}{2}

This is correct

Posted by

Vakul

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