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Let   a  =   m + ni

Let  b conjugate =     p  - qi  

So

a* (b conjugate)  =   (m + ni) (p - qi)  =  mp + (np - mq)i  + nq    =  (mp + nq) + (np - mq) i =   -1 + 5i 

mp + nq  =  -1

np - mq  = 5

And

a conjugate  =  m - ni

b =  p + qi

So

(a conjugate) * b  =    ( m - ni) ( p + qi)  = mp + (mq - np)i + nq  = ( mp + nq)  + (mq - np)i 

So

(mp + nq)  =  -1

(mq - np)  =  -5

So

(a conjugate)  * b   =    -1  - 5i

The complex numbers a and b satisfy  \(a\overline{b} = -1 + 5i\).
Find \(\overline{a} b\).

\(\begin{array}{|rcll|} \hline a\overline{b} &=& -1 + 5i \\ \overline{a\overline{b}} &=& \overline{-1 + 5i} \quad | \quad \overline{a\overline{b}} = \overline{a}\overline{\overline{b}} \\ \overline{a}\overline{\overline{b}} &=& \overline{-1 + 5i} \quad | \quad \overline{a}\overline{\overline{b}} = \overline{a} b \\ \overline{a} b &=& \overline{-1 + 5i} \quad | \quad \overline{-1 + 5i} = -1 - 5i \\ \mathbf{\overline{a} b} &=& \mathbf{ -1 - 5i } \\ \hline \end{array}\)