If (z - α)/(z + α)(α ∈ R) is a purely imaginary number and |z| = 2...

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

$⇒ \dfrac{z - \alpha}{z + \alpha} = - \left(\dfrac{z - \alpha}{z + \alpha}\right) ⇒ \dfrac{z - \alpha}{z + \alpha} = - \left(\dfrac{\overset{-}{z} - \alpha}{\overset{-}{z} + \alpha}\right)$

$⇒ \left(z - \alpha\right)\left(\overset{-}{z} + \alpha\right) = \left(z + \alpha\right)\left(\overset{-}{z} - \alpha\right)$

$⇒ z\overset{-}{z} + \alphaz - \alpha\overset{-}{z} - {\alpha}^{2} = - \left(z\overset{-}{z} - \alphaz + \alpha\overset{-}{z} - {\alpha}^{2}\right)$

$⇒ {\left|z\right|}^{2} = {\alpha}^{2}$

$⇒ {\alpha}^{2} = 4\left[∵ \left|z\right| = 2\right]$

$⇒ \alpha = \pm 2$