SATHEE: Complex Numbers 2 Question 4

Complex Numbers 2 Question 4

4. If $\frac{z - α}{z + α} (α \in R)$ is a purely imaginary number and $| z | = 2$ , then a value of $α$ is

(2019 Main, 12 Jan I)

(a) $\sqrt{2}$

(b) $\frac{1}{2}$

(c) 1

(d) 2

Show Answer

Answer:

Correct Answer: 4. (d)

Solution:

Since, the complex number $\frac{z - α}{z + α} (α \in R)$ is purely imaginary number, therefore

$\begin{array}{rrr} \frac{z - α}{z + α} + \frac{\bar{z} - α}{\bar{z} + α} = 0 & [∵ α \in R] \\ \Rightarrow & z \bar{z} - α \bar{z} + α z - α^{2} + z \bar{z} - α z + α \bar{z} - α^{2} = 0 \\ \Rightarrow & 2 | z |^{2} - 2 α^{2} = 0 & [∵ z \bar{z} = | z |^{2}] \\ \Rightarrow & α^{2} = | z |^{2} = 4 & [| z | = 2 given] \\ \Rightarrow & α = \pm 2 \end{array}$

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Complex Numbers 2 Question 4

4. If z−αz+α(α∈R) is a purely imaginary number and |z|=2, then a value of α is

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4. If $\frac{z - α}{z + α} (α \in R)$ is a purely imaginary number and $| z | = 2$ , then a value of $α$ is