Let `(z-alpha)/(z+alpha)` is purely imaginary and `|z|=2, alphaepsilonR` then `alpha` is equal to

Options

A. `sqrt(2)`
B. `1/2`
C. 1
D. 2

Answer

D

Explanation

Since the complex number `(z-a)/(z+a)(alpha ne R)`j is purely imaginary number therefore
`(z-alpha)/(z+alpha)+(barz-alpha)/(bar z + alpha)=0 " "[ therefore alpha in R]`
`rArr zbarz-abarz+az-alpha^2+zbarz-alphabarz-alpha^2=0`
`rArr 2|z|^2-2alpha^2=0" "[because zbarz=|z|^2]`
`rArr alpha^2=|z|^2=4`
`rArr alpha=pm2`