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The equation of the line joining the complex numbers -5+4i and 7+2i can be expressed in the form

az + b \overline{z} = 38
for some complex numbers a and b. Find the product ab.

 Jun 5, 2022
 #1
avatar+26321 
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The equation of the line joining the complex numbers
\(-5+4i\) and \(7+2i\) can be expressed in the form
\(az + b \overline{z} = 38\)
for some complex numbers a and b. Find the product ab
.


\(\text{Let $z_1=-5+4i$ and $\overline{z_1}=-5-4i$} \\ \text{Let $z_2=7+2i$ and $\overline{z_2}=7-2i$} \\ \text{Let $az_1 + b \overline{z_1} = 38$} \\ \text{Let $az_2 + b \overline{z_2} = 38$}\)

 

\(\begin{array}{|rcll|} \hline az_2 + b \overline{z_2} &=& 38 \\ az_2 &=& 38- b \overline{z_2} \\ \mathbf{a} &=& \mathbf{\dfrac{38- b \overline{z_2}}{z_2}} \\ \hline az_1 + b \overline{z_1} &=& 38 \\\\ \mathbf{\dfrac{38- b \overline{z_2}}{z_2}}z_1 + b \overline{z_1} &=& 38 \quad | \quad *z_2 \\\\ (38- b \overline{z_2})z_1 + b \overline{z_1}z_2 &=& 38z_2 \\ 38z_1- b \overline{z_2}z_1+ b \overline{z_1}z_2 &=& 38z_2 \\ b(\overline{z_1}z_2-\overline{z_2}z_1) &=& 38(z_2-z_1) \\\\ \mathbf{b} &=& \mathbf{ \dfrac{38(z_2-z_1)}{ (\overline{z_1}z_2-\overline{z_2}z_1) }} \\ \hline \end{array} \begin{array}{|rcll|} \hline && \overline{z_1}z_2-\overline{z_2}z_1\\ &=& (-5-4i)(7+2i)-(7-2i)(-5+4i) \\ &=& \mathbf{-76i} \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline \mathbf{b} &=& \mathbf{ \dfrac{38(z_2-z_1)}{\overline{z_1}z_2-\overline{z_2}z_1}} \\\\ &=& \dfrac{38\Big(7+2i-(-5+4i) \Big)}{-76i} \\\\ &=& \dfrac{38(7+2i+5-4i)}{-76i} \\\\ &=& \dfrac{7+2i+5-4i}{-2i} \\\\ &=& \dfrac{12-2i}{-2i} \\\\ &=& \dfrac{-6+i}{i} \\\\ &=& \dfrac{-6}{i}+1 \\\\ &=& \dfrac{-6i}{i^2}+1 \\\\ &=& \dfrac{-6i}{-1}+1 \\\\ &=& 6i+1 \\ \mathbf{b} &=& \mathbf{1+6i} \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline az_2 + b \overline{z_2} &=& 38 \\\\ az_2 + \mathbf{ \dfrac{38(z_2-z_1)}{(\overline{z_1}z_2-\overline{z_2}z_1)}} \overline{z_2} &=& 38 \\ az_2 &=& 38-\dfrac{38(z_2-z_1)\overline{z_2}}{(\overline{z_1}z_2-\overline{z_2}z_1)} \quad | \quad : z_2 \\\\ a &=& \dfrac{38}{z_2}-\dfrac{38(z_2-z_1)\overline{z_2}}{(\overline{z_1}z_2-\overline{z_2}z_1)z_2} \\\\ a &=& \dfrac{38(\overline{z_1}z_2-\overline{z_2}z_1)-38(z_2-z_1)\overline{z_2}}{z_2(\overline{z_1}z_2-\overline{z_2}z_1)} \\\\ a &=& \dfrac{38 \Big(\overline{z_1}z_2-\overline{z_2}z_1-(z_2-z_1)\overline{z_2} \Big) } {z_2(\overline{z_1}z_2-\overline{z_2}z_1)} \\\\ a &=& \dfrac{38 (\overline{z_1}z_2-\overline{z_2}z_1-z_2\overline{z_2}+z_1\overline{z_2}) } {z_2(\overline{z_1}z_2-\overline{z_2}z_1)} \\\\ a &=& \dfrac{38 (\overline{z_1}z_2-z_2\overline{z_2}) } {z_2(\overline{z_1}z_2-\overline{z_2}z_1)} \\\\ a &=& \dfrac{38z_2 (\overline{z_1}-\overline{z_2}) } {z_2(\overline{z_1}z_2-\overline{z_2}z_1)} \\\\ \mathbf{a} &=& \mathbf{ \dfrac{38(\overline{z_1}-\overline{z_2}) } {(\overline{z_1}z_2-\overline{z_2}z_1)} } \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline \mathbf{a} &=& \mathbf{ \dfrac{38(\overline{z_1}-\overline{z_2}) } {(\overline{z_1}z_2-\overline{z_2}z_1)} } \\\\ &=& \dfrac{38\Big(-5-4i-(7-2i) \Big)}{-76i} \\\\ &=& \dfrac{38(-7+2i-5-4i)}{-76i} \\\\ &=& \dfrac{-7+2i-5-4i}{-2i} \\\\ &=& \dfrac{-12-2i}{-2i} \\\\ &=& \dfrac{6+i}{i} \\\\ &=& \dfrac{6}{i}+1 \\\\ &=& \dfrac{6i}{i^2}+1 \\\\ &=& \dfrac{6i}{-1}+1 \\\\ &=& -6i+1 \\ \mathbf{a} &=& \mathbf{1-6i} \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline ab &=& (1+6i)(1-6i) \\ ab &=& 1+36 \\ \mathbf{ab} &=& \mathbf{37} \\ \hline \end{array}\)

 

laugh

 Jun 6, 2022

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