Consider the line with equation
\((2-i)z + (2+i)\overline{z} = 20.\)
For each complex number in the following list,
\(2, 4+2i, 3i, 1-4i, 5, 3, 10i,\)
figure out whether each one is on the line, then enter "yes" or "no" in the blank corresponding to each option.
+26367 Consider the line with equation
\(\large (2-i)z + (2+i)\overline{z} = 20. \)
For each complex number in the following list,
\(\large 2,\ 4+2i,\ 3i,\ 1-4i,\ 5,\ 3,\ 10i,\)
figure out whether each one is on the line.
\(\text{Let $z = a+bi$} \\ \text{Let $\overline{z} = a-bi$}\)
\(\begin{array}{|rcll|} \hline (2-i)z + (2+i)\overline{z} &=& 20 \quad & | \quad z = a+bi, \ \overline{z} = a-bi \\ (2-i)(a+bi) + (2+i)(a-bi) &=& 20 \\ 2a+2bi-ia-bi^2 +2a -2bi +ia -bi^2 &=& 20 \\ 2a-bi^2 +2a -bi^2 &=& 20 \\ 4a-2bi^2 &=& 20 \quad & | \quad i^2 = -1 \\ 4a+2b &=& 20 \quad & | \quad : 2 \\ \mathbf{2a+b} &\mathbf{=}& \mathbf{10} \\ \hline \end{array}\)
\(\begin{array}{|l|l|c|l|c|} \hline &\text{list } z = a+bi:&& \mathbf{2a+b = 10} & \text{on the line} \\ \hline 1) & 2: & a= 2 & 2\cdot 2 + 0 \ne 10 \\ & & b= 0 \\\\ \hline 2) & 4+2i: & a= 4 & 2\cdot 4 + 2 \mathbf{= 10} & \checkmark \\ & & b= 2 \\\\ \hline 3) & 3i: & a= 0 & 2\cdot 0 + 3 \ne 10 \\ & & b= 3 \\\\ \hline 4) & 1-4i: & a= 1 & 2\cdot 1 -4 \ne 10 \\ & & b= -4 \\\\ \hline 5) & 5: & a= 5 & 2\cdot 5 + 0 \mathbf{= 10} & \checkmark \\ & & b= 0 \\\\ \hline 6) & 3: & a= 3 & 2\cdot 3 + 0 \ne 10 \\ & & b= 0 \\\\ \hline 7) & 10i: & a= 0 & 2\cdot 0 + 10 \mathbf{= 10} & \checkmark \\ & & b= 10 \\\\ \hline \end{array} \)
