Question:
If $a>0$ and $z=\frac{(1+i)^{2}}{a-i}$, has magnitude $\sqrt{\frac{2}{5}}$, then $\bar{z}$ is equal to :
Correct Option: , 3
Solution:
Given $a>0$
$z=\frac{(1+i)^{2}}{a-i}=\frac{2 i(a+i)}{a^{2}+1}$
Also $|z|=\sqrt{\frac{2}{5}} \Rightarrow \frac{2}{\sqrt{a^{2}+1}}=\sqrt{\frac{2}{5}} \Rightarrow a=3$
So $\bar{z}=\frac{-2 i(3-i)}{10}=\frac{-1-3 i}{5}$
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