If w w z 1 z is purely real where w = + i , 0 and z 1 , then the set of the values of z is [2006]

Question

If $\frac{w - \bar{w} z}{1 - z}$ is purely real where $w = α + i β, β \neq 0$ and $z \neq 1$ , then the set of the values of $z$ is [2006]

Smart Achievers · Accepted Answer

(4)

$∵$ $\frac{w - \bar{w} z}{1 - z} is purely real$

$∴$ $\bar{(\frac{w - \bar{w} z}{1 - z})} = (\frac{w - \bar{w} z}{1 - z}) \Rightarrow \frac{\bar{w} - w \bar{z}}{1 - \bar{z}} = \frac{w - \bar{w} z}{1 - z}$

$\Rightarrow \bar{w} - w \bar{z} - \bar{w} z + w z \bar{z} = w - \bar{w} z - w \bar{z} + \bar{w} z \bar{z}$

$\Rightarrow w - \bar{w} = (w - \bar{w}) | z |^{2}$

$\Rightarrow | z |^{2} = 1$ $(∵ w = α + i β and β \neq 0)$

$\Rightarrow$ $| z |$ $= 1 and also given that z \neq 1$

$∴$ $The required set is {z : | z | = 1, z \neq 1}$

Solution

Q.
If $\frac{w - \bar{w} z}{1 - z}$ is purely real where $w = α + i β, β \neq 0$ and $z \neq 1$ , then the set of the values of $z$ is [2006]

1 ${z : | z | = 1}$

2 ${z : z = \bar{z}}$

3 ${z : z \neq 1}$

4 ${z : | z | = 1, z \neq 1}$

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Solution

Q. If w-w¯z1-z is purely real where w=α+iβ, β≠0 and z≠1, then the set of the values of z is [2006]

1 {z:|z|=1}

2 {z:z=z¯}

3 {z:z≠1}

4 {z:|z|=1, z≠1}