If z   -   α z   +   α α ∈ R is a purely imaginary number and z = 2 , then a value of α is :

Question

If z &#160; - &#160; &#945; z &#160; + &#160; &#945; &#945; &#8712; R is a purely imaginary number and z = 2 , then a value of &#945; is :

Aicharya · Accepted Answer

If a complex number is purely imaginary, then it must be equal to negative of its conjugate.

$\Rightarrow \frac{z - α}{z + α} = - (\frac{z - α}{z + α}) \Rightarrow \frac{z - α}{z + α} = - (\frac{\bar{z} - α}{\bar{z} + α})$

$\Rightarrow (z - α) (\bar{z} + α) = (z + α) (\bar{z} - α)$

$\Rightarrow z \bar{z} + α z - α \bar{z} - α^{2} = - (z \bar{z} - α z + α \bar{z} - α^{2})$

$\Rightarrow {|z|}^{2} = α^{2}$

$\Rightarrow α^{2} = 4 [∵ |z| = 2]$

$\Rightarrow α = \pm 2$

If z   -   α z   +   α α ∈ R is a purely imaginary number and z = 2…