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Consider the line with equation \[(2-i)z + (2+i)\overline{z} = 20.\]
Where does this line intersect the real axis?

 Jan 18, 2019
 #1
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Consider the line with equation

\( (2-i)z + (2+i)\overline{z} = 20. \)

 \[(2-i)z + (2+i)\overline{z} = 20.\]
Where does this line intersect the real axis?

 

On the real axis:  \(z = \overline z\)

\(\begin{array}{|rcll|} \hline (2-i)z + (2+i)\overline{z} &=& 20 \quad & | \quad z = \overline z \\ (2-i)z + (2+i)z &=& 20 \\ z\Big((2-i)+(2+i)\Big) &=& 20 \\ z(2-i+2+i) &=& 20 \\ 4z &=& 20 \\ \mathbf{ z } & \mathbf{=} & \mathbf{5} \\ \hline \end{array}\)

 

The line intersect the real axis at 5.

 

laugh

 Jan 18, 2019

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