SATHEE: Complex Numbers 2 Question 12

Complex Numbers 2 Question 12

12.

If $w = α + i β$ , where $β \neq 0$ and $z \neq 1$ , satisfies the condition that $(\frac{w - \bar{w} z}{1 - z})$ is purely real, then the set of values of $z$ is

$(2006, 3 M)$

(a) $| z | = 1, z \neq 2$

(b) $| z | = 1$ and $z \neq 1$

(d) None of these

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Answer:

Correct Answer: 12. (b)

Solution:

Let $z_{1} = \frac{w - \bar{w} z}{1 - z}$ be purely real $\Rightarrow z_{1} = {\bar{z}}_{1}$

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

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

$\Rightarrow (w - \bar{w}) + (\bar{w} - w) | z |^{2} = 0$

$\Rightarrow (w - \bar{w}) (1 - | z |^{2}) = 0$

$\Rightarrow | z |^{2} = 1$

[as $w - \bar{w} \neq 0$ , since $β \neq 0]$

$\Rightarrow | z | = 1$ and $z \neq 1$

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Complex Numbers 2 Question 12

12.

Answer:

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