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Let z and w be complex numbers satisfying |z| = 5, |w| = 2, and z(overline{w}) = 6+8i. 

Find the value of |z+w|^2, |zw|^2, |z-w|^2,  |frac{z}{w}|^2 .

If any of these cannot be uniquely determined from the information given, write not possible.

 May 10, 2019
 #1
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Let \(z\) and \(w\) be complex numbers satisfying \(|z| = 5\), \(|w| = 2\), and \(z\overline{w} = 6+8i\)

Find the value of \(|z+w|^2\), \(|zw|^2\), \(|z-w|^2\),  \(\left|\dfrac{z}{w}\right|^2\) .

 

\(\begin{array}{|rcll|} \hline && \mathbf{|z+w|^2} \\ &=& |z|^2+|w|^2+2*\Re{(z\overline{w})} \\ &=& 5^2+2^2+2*6 \quad | \quad \Re{(z\overline{w})} = \Re{(6+8i)} = 6 \\ &=& \mathbf{41} \\ \hline && \mathbf{|zw|^2} \\ &=&|z|^2|w|^2 \\ &=& 5^2*2^2 \\ &=& 25*4 \\ &=& \mathbf{100} \\ \hline && \mathbf{|z-w|^2} \\ &=& |z|^2+|w|^2-2*\Re{(z\overline{w})} \\ &=& 5^2+2^2-2*6 \quad | \quad \Re{(z\overline{w})} = \Re{(6+8i)} = 6 \\ &=& \mathbf{17} \\ \hline && \mathbf{\left|\dfrac{z}{w}\right|^2} \\ &=& \dfrac{|z|^2}{|w|^2} \\ &=& \dfrac{5^2}{2^2} \\ &=& \dfrac{25}{4} \\ &=& \mathbf{6.25} \\ \hline \end{array}\)

 

 

laugh

 May 10, 2019

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