If `k>0`, `|z|=|w|=k`, and `alpha=(z-bar w)/(k^2+zbar(w))`, then `Re(alpha)` (A) 0 (B) `k/2` (C) `k` (D) None of these

Options

A. 0
B. `k//2`
C. k
D. none of these

Answer

A

Explanation

We have,
`|z|=|omega|=k`
`rArr |z|^(2)=|omega|^(2)=k^(2) rArr zbarz=k^(2) rArr omegabar(omega)=k^(2)`
Now, `alpha=(z-baromega)/(k^(2)+zbaromega)`
`rArr baralpha = (barz-omega)/(k^(2)+barzomega) = (k^(2)/z-k^(2)/baromega)/(k^(2)+k^(2)/z xx k^(2)/baromega) =- alpha`
`rArr alpha+baralpha=0 rArr "Re"(alpha)=0`